combination - a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter

Formula:_{n}P_{r}= n!/r!(n - r)!

11 combination of 3 times 6 combination of 3

11 combination of 3 times 6 combination of 3
[URL='https://www.mathcelebrity.com/permutation.php?num=11&den=3&pl=Combinations']11 combination of 3[/URL] = 165
[URL='https://www.mathcelebrity.com/permutation.php?num=6&den=3&pl=Combinations']6 combination of 3[/URL] = 20
11 combination of 3 times 6 combination of 3 = 165 * 20
11 combination of 3 times 6 combination of 3 = [B]3300[/B]

3 salads, 4 main dishes, and 2 desserts

3 salads, 4 main dishes, and 2 desserts
Total meal combinations are found by multiplying each salad, main dish, and dessert using the fundamental rule of counting.
The fundamental rule of counting states, if there are a ways of doing one thing, b ways of doing another thing, and c ways of doing another thing, than the total combinations of all the ways are found by a * b * c.
With 3 salads, 4 main dishes, and 2 desserts, our total meal combinations are:
3 * 4 * 2 = [B]24 different meal combinations.[/B]

A 3-digit security code can use the numbers 0-9. How many possible combinations are there if the num

A 3-digit security code can use the numbers 0-9. How many possible combinations are there if the numbers can be repeated
[0-9] * [0-9] * [0-9]
10 * 10 * 10 = [B]1,000 combinations[/B]

A bag contains 8 marbles of different colors. How many ways unique ways or orders can you select 3 m

A bag contains 8 marbles of different colors. How many ways unique ways or orders can you select 3 marbles?
We want the combinations formula, 8 choose 3, or 8C3.
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']Type 8 C 3 into our search engine and we get:[/URL]
[B]56 unique ways[/B]

A catering service offers 3 appetizers, 6 main courses, and 4 desserts. A customer is to select 2 ap

A catering service offers 3 appetizers, 6 main courses, and 4 desserts. A customer is to select 2 appetizers, 3 main courses, and 3 desserts for a banquet. In how many ways can this be done?
We use the combinations formula, and since each event is independent of the others, we multiply:
2 appetizers, 3 main courses, and 3 desserts = [URL='https://www.mathcelebrity.com/permutation.php?num=3&den=2&pl=Combinations']3C2[/URL] * [URL='https://www.mathcelebrity.com/permutation.php?num=6&den=3&pl=Combinations']6C3[/URL] * [URL='https://www.mathcelebrity.com/permutation.php?num=4&den=3&pl=Combinations']4C3[/URL]
2 appetizers, 3 main courses, and 3 desserts = 3 * 20 * 4
2 appetizers, 3 main courses, and 3 desserts = [B]240[/B]

A catering service offers 4 appetizers, 12 main courses, and 9 desserts. A customer is to select 3 a

A catering service offers 4 appetizers, 12 main courses, and 9 desserts. A customer is to select 3 appetizers, 10 main courses, and 5 desserts for a banquet. In how many ways can this be done?
We use combinations, so we have:
[LIST]
[*][URL='https://www.mathcelebrity.com/permutation.php?num=4&den=3&pl=Combinations']4C3 appetizers[/URL] = 4
[*][URL='https://www.mathcelebrity.com/permutation.php?num=12&den=10&pl=Combinations']12C10 main courses[/URL] = 66
[*][URL='https://www.mathcelebrity.com/permutation.php?num=9&den=5&pl=Combinations']9C5 desserts[/URL] = 126
[/LIST]
We multiply each of these together to get our total combinations:
4 * 66 * 126 = [B]33,264[/B]

A combination lock open with the correct 4 letter code. Each wheel roared through letters A-L. How m

A combination lock open with the correct 4 letter code. Each wheel roared through letters A-L. How many different 4 letter codes are possible
A-L = 12 letters
Possible combinations is found by:
12 * 12 * 12 * 12 = [B]20,736 combinations[/B]

A committee consisting of 4 faculty members and 5 students is to be formed. Every committee position

A committee consisting of 4 faculty members and 5 students is to be formed. Every committee position has the same duties and voting rights. There are 12 faculty members and 15 students eligible to serve on the committee. In how many ways can the committee be formed?
We'll use combinations, so we have:
[LIST]
[*][URL='https://www.mathcelebrity.com/permutation.php?num=12&den=4&pl=Combinations']12 faculty members choose 4 faculty members --> 12 C 4[/URL] = 495
[*][URL='https://www.mathcelebrity.com/permutation.php?num=15&den=5&pl=Combinations']15 students choose 5 students --> 15 C 5[/URL] = 3,003
[/LIST]
To get the total committees, we multiply the total faculty member choices by the total student choices:
Total committees = total faculty members * total students
Total committees = 495 * 3,003
Total committees = [B]1,486,485[/B]

A committee of 6 students are being selected from a class of 10 girls and 8 boys. How many committee

A committee of 6 students are being selected from a class of 10 girls and 8 boys. How many committees are possible if three must be girls and 3 must be boys?
We want combinations. How many ways can we choose 3 boys from 8 boys:
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']8 choose 3[/URL] = 56
We want combinations. How many ways can we choose 3 girls from 10 girls:
[URL='https://www.mathcelebrity.com/permutation.php?num=10&den=3&pl=Combinations']10 choose 3[/URL] = 120
Our total choices are found by multiplying each event:
Total committees = (8 boys choose 3) * (10 girls choose 3)
Total committees = 56 * 120
Total committees = [B]6,720[/B]

A family has 4 children. Give the sample space in regards to the genders of the children

A family has 4 children. Give the sample space in regards to the genders of the children.
Children can either be male or female.
Therefore, the sample space is 2 * 2 * 2 * 2 = 16 possible combinations.
[LIST=1]
[*]MMMM
[*]MMMF
[*]MMFM
[*]MFMM
[*]FMMM
[*]MMFF
[*]MFFM
[*]FFMM
[*]MFMF
[*]FMFM
[*]MMMF
[*]FMMM
[*]FFFM
[*]MFFF
[*]FMMF
[*]FFFF
[/LIST]

A jar contains 80 nickels and dimes worth $6.40. How many of each kind of coin are in the jar?

A jar contains 80 nickels and dimes worth $6.40. How many of each kind of coin are in the jar?
Using our [URL='http://www.mathcelebrity.com/coin-word-problem.php?coinvalue=6.40&cointot=80&coin1=nickels&coin2=dimes&pl=Calculate+Coin+Quantities']coin combination word problem calculator[/URL], we get:
[LIST]
[*][B]48 dimes[/B]
[*][B]32 nickels[/B]
[/LIST]

a licence plate that has 3 numbers from 0 to 9 followed by 2 letters

a licence plate that has 3 numbers from 0 to 9 followed by 2 letters
How many license plate combinations can we form?
We multiply as follows:
[LIST]
[*][0-9] = 10 possible digits (D)
[*]A-Z = 26 possible letters (L)
[/LIST]
The problem asks for this:
DDDLL
So we have:
10 * 10 * 10 * 26 * 26 = [B]676,000[/B] plates

a license plate has 3 letters followed by 4 numbers

a license plate has 3 letters followed by 4 numbers
There are 26 letters A-Z and 10 numbers 0-9. So we have:
26 * 26 * 26 * 10 * 10 * 10 * 10
[B]175,760,000 different license plate combinations[/B]

A license plate is made up of 2 letter and 3 single digit numbers

A license plate is made up of 2 letter and 3 single digit numbers.
There are 26 letters (A-Z). And there are 10 single digit numbers [0-9]. So our total combinations are:
Letter - Letter - Number - Number - Number
26 * 26 * 10 * 10 * 10 = [B]676,000[/B]

A local college classifies its students by major, year (Freshman, Sophomore, Junior, Senior) and sex

A local college classifies its students by major, year (Freshman, Sophomore, Junior, Senior) and sex (M, F). If the college offers 20 majors, how many combinations are possible?
We have 20 majors, 4 grade levels, and 2 sexes.
The total combinations = 20 * 4 * 2 = [B]160[/B]

A music camp with 50 students decided to break the students into barbershop quartets to see which co

A music camp with 50 students decided to break the students into barbershop quartets to see which combination of four students sounded the best. How many different barbershop quartets can be made with 50 students so that each possible combinations of four is tried?
We want 50 combinations of 4.
[URL='https://www.mathcelebrity.com/permutation.php?num=50&den=4&pl=Combinations']50C4 [/URL]= 230,300

A number cube is rolled and a coin is tossed. The number cube and the coin are fair. What is the pro

A number cube is rolled and a coin is tossed. The number cube and the coin are fair. What is the probability that the number rolled is greater than 3 and the coin toss is heads? Write your answer as a fraction in simplest form
Let's review the vitals of this question:
[LIST]
[*]The probability of heads on a fair coin is 1/2.
[*]On a fair die, greater than 3 means either 4, 5, or 6. Any die roll face is a 1/6 probability.
[*]So we have a combination of outcomes below:
[/LIST]
Outcomes
[LIST=1]
[*]Heads and 4
[*]Heads and 5
[*]Heads and 6
[/LIST]
For each of the outcomes, we assign a probability. Since the coin flip and die roll are independent, we multiply the probabilities:
[LIST=1]
[*]P(Heads and 4) = 1/2 * 1/6 = 1/12
[*]P(Heads and 5) = 1/2 * 1/6 = 1/12
[*]P(Heads and 6) = 1/2 * 1/6 = 1/12
[/LIST]
Since we want any of those events, we add all three probabilities
1/12 + 1/12 + 1/12 = 3/12
This fraction is not simplified. S[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F12&frac2=3%2F8&pl=Simplify']o we type this fraction into our search engine, and choose Simplify[/URL].
We get a probability of [B]1/4[/B].
By the way, if you need a decimal answer or percentage answer instead of a fraction, we type in the following phrase into our search engine:
[URL='https://www.mathcelebrity.com/perc.php?num=1&den=4&pcheck=1&num1=+16&pct1=+80&pct2=+35&den1=+90&pct=+82&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']1/4 to decimal[/URL]
Alternative Answers:
[LIST]
[*]For a decimal, we get [B]0.25[/B]
[*]For a percentage, we get [B]25%[/B]
[/LIST]

A pollster selected 4 of 7 people. How many different groups of 4 are possible?

A pollster selected 4 of 7 people. How many different groups of 4 are possible?
We want to use the combinations formula. [URL='https://www.mathcelebrity.com/permutation.php?num=7&den=4&pl=Combinations']So we type 7C4 into our search engine[/URL]. This is also known as 7 choose 4.
We get [B]35[/B] different groups.

A professor wanted to test all possible pairwise comparisons among six means. How many comparisons d

A professor wanted to test all possible pairwise comparisons among six means. How many comparisons did he need to compare?
a. 5
b. 6
c. 10
d. 15
[B]d. 15[/B] using our [URL='http://www.mathcelebrity.com/permutation.php?num=6&den=2&pl=Combinations']combinations calculator[/URL]

A restaurant has 8 pizza toppings to choose from. How many different 2 topping pizzas are possible?

A restaurant has 8 pizza toppings to choose from. How many different 2 topping pizzas are possible?
We want 8 combinations of 2, denoted as 8 C 2, or 8 choose 2.
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=2&pl=Combinations']Typing 8 C 2 into the search engine[/URL], we get [B]28[/B] different 2 topping pizzas

A restaurant offers a special pizza with any 5 toppings. If the restaurant has 17 topping from which

A restaurant offers a special pizza with any 5 toppings. If the restaurant has 17 topping from which to choose, how many different special pizzas are possible?
We have 17 choose 5, or 17C5.
[URL='https://www.mathcelebrity.com/permutation.php?num=17&den=5&pl=Combinations']Type this into the search engine[/URL], and we get [B]6,188[/B] different special pizzas available.

A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 6 f

A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 6 freshmen, 9 sophomores, 7 juniors, and 7 seniors are eligible to be on the committee, in how many ways can the committee be chosen?
We want combinations for freshmen, sophomores, juniors, and seniors.
[LIST]
[*]Freshmen choices: [URL='https://www.mathcelebrity.com/permutation.php?num=6&den=2&pl=Combinations']6 C 2[/URL] = 15
[*]Sophomore choices: [URL='https://www.mathcelebrity.com/permutation.php?num=9&den=3&pl=Combinations']9 C 3[/URL] = 84
[*]Junior choices: [URL='https://www.mathcelebrity.com/permutation.php?num=7&den=4&pl=Combinations']7 C 4[/URL] = 35
[*]Senior choices: [URL='https://www.mathcelebrity.com/permutation.php?num=7&den=5&pl=Combinations']7 C 5 [/URL]= 21
[/LIST]
The number of committees we can choose is the product of combinations of freshmen, sophomores, juniors, and seniors.
Total Committees = Freshmen choices * Sophomore choices * Junior choices * Senior choices
Total Committees = 15 * 84 * 35 * 21
Total Committees = [B]926,100[/B]

A small acting club has 8 members. How many different 2 member groups are possible?

A small acting club has 8 members. How many different 2 member groups are possible?
We want 8C2
Using our [URL='http://www.mathcelebrity.com/permutation.php?num=8&den=2&pl=Combinations']combinations calculator[/URL], we get:
[B]28 possible groups[/B]

A used book store also started selling used CDs and videos. In the first week, the store sold a comb

A used book store also started selling used CDs and videos. In the first week, the store sold a combination of 40 CDs and videos. They charged $4 per CD and $6 per video and the total sales were $180. Determine the total number of CDs and videos sold
Let c be the number of CDs sold, and v be the number of videos sold. We're given 2 equations:
[LIST=1]
[*]c + v = 40
[*]4c + 6v = 180
[/LIST]
You can solve this system of equations three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter what method we choose, we get [B]c = 30, v = 10[/B].
Now let's check our work for both given equations for c = 30 and v = 10:
[LIST=1]
[*]30 + 10 = 40 <-- This checks out
[*]4c + 6v = 180 --> 4(30) + 6(10) --> 120 + 60 = 180 <-- This checks out
[/LIST]

A used book store also started selling used CDs and videos. In the first week, the store sold a comb

A used book store also started selling used CDs and videos. In the first week, the store sold a combination of 40 CDs and videos. They charged $4 per CD and $6 per video and the total sales were $180. Determine the total number of CDs and videos sold.
Let the number of cd's be c and number of videos be v. We're given two equations:
[LIST=1]
[*]c + v = 40
[*]4c + 6v = 180
[/LIST]
We can solve this system of equations using 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[B]c = 30
v = 10[/B]

Abbey knew that the combination for her locker had the numbers 36, 12, 8, and 40, but she couldn't r

Abbey knew that the combination for her locker had the numbers 36, 12, 8, and 40, but she couldn't remember the right order of the numbers. How many different possibilities are there for the lock combination using the four numbers?
First number could be 4 choices, then 3, then 2, then 1. So we have:
4! = 4 x 3 x 2 x 1 = [B]24 possibilities[/B]

CCP Stat Club is sending a delegate of 2 members to attend the 2015 Joint Statistical Meeting in Sea

CCP Stat Club is sending a delegate of 2 members to attend the 2015 Joint Statistical Meeting in Seattle. There are 10 qualified candidates. How many different ways can the delegate be selected?
10C2 = [B]45[/B] shown on our [URL='http://www.mathcelebrity.com/permutation.php?num=10&den=2&pl=Combinations']Combinations Calculator[/URL]

Chris has 6 cds that he is going to give away. He let his best friend choose 2 of the 6 cds. How man

Chris has 6 cds that he is going to give away. He let his best friend choose 2 of the 6 cds. How many different groups of 2cds are possible?
We want 6 choose 2 using combinations. We use combinations because the problem states the word [I]different[/I].
[URL='https://www.mathcelebrity.com/permutation.php?num=6&den=2&pl=Combinations']6 C 2[/URL] using our calculator is [B]15[/B].

Coin Combinations

Free Coin Combinations Calculator - Given a selection of coins and an amount, this determines the least amount of coins needed to reach that total.

Combination with Variable

Free Combination with Variable Calculator - Calculates the following:

Solves for r given n and the combination value.

Solves for n given r and the combination value

Solves for r given n and the combination value.

Solves for n given r and the combination value

Combinations with Replacement

Free Combinations with Replacement Calculator - Calculates the following:

How many combinations can we have from a sample of r elements from a set of n distinct objects where order does matter and replacements are not allowed?

How many combinations can we have from a sample of r elements from a set of n distinct objects where order does matter and replacements are not allowed?

Committees of 4 men 5 women form a group of 11 men and 11 women.

Committees of 4 men 5 women form a group of 11 men and 11 women.
We want combinations.
4 men from 11 men is the combination 11C4. [URL='https://www.mathcelebrity.com/permutation.php?num=11&den=4&pl=Combinations']Using our combinations calculator[/URL], we get:
11C4 = 330
5 women from 11 women is the combination 11C5. [URL='https://www.mathcelebrity.com/permutation.php?num=11&den=5&pl=Combinations']Using our combinations calculator[/URL], we get:
11C5 = 462
We multiply the committee of men times the committee of women:
11C4 * 11C5 = 330 * 432
11C4 * 11C5 = [B]142,560[/B]

Eight people have volunteered for a secret mission that requires only 3 people. How many different c

Eight people have volunteered for a secret mission that requires only 3 people. How many different combinations are possible?
We want 8 combinations of 3
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']8C3 [/URL]= 56

Evaluate the expression (8C3) (7C6)

Evaluate the expression (8C3) (7C6)
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']8C3[/URL] = 56
[URL='https://www.mathcelebrity.com/permutation.php?num=7&den=6&pl=Combinations']7C6[/URL] = 7
(8C3) (7C6) = 56 * 7
(8C3) (7C6) = [B]392[/B]

Find the number of combinations and the number of permutations for 10 objects taken 6 at a time

Find the number of combinations and the number of permutations for 10 objects taken 6 at a time
[LIST]
[*]Combinations is written as 10 C 6. Using our [URL='http://www.mathcelebrity.com/permutation.php?num=10&den=6&pl=Combinations']combinations calculator[/URL], we get [B]210[/B].
[*]Permutations is written as 10 P 6. Using our [URL='http://www.mathcelebrity.com/permutation.php?num=10&den=6&pl=Permutations']permutations calculator[/URL], we get [B]151,200[/B].
[/LIST]

Five players are going to be picked to start a basketball game. If there are 13 players on the team,

Five players are going to be picked to start a basketball game. If there are 13 players on the team, how many different combinations of 5 starting players can be made?
We want 13 combinations of 5
[URL='https://www.mathcelebrity.com/permutation.php?num=13&den=5&pl=Combinations']13C5[/URL] = [B]1287[/B]

From 9 names on a ballot, a committee of 4 will be elected to attend a political national convention

From 9 names on a ballot, a committee of 4 will be elected to attend a political national convention. How many different committees are possible?
We want unique combinations, so we have 9 choose 4, or 9C4.
[URL='https://www.mathcelebrity.com/permutation.php?num=9&den=4&pl=Combinations']Typing this into the search engine[/URL], we get:
9C4 = [B]126 different committees[/B]

From a group of 10 men and 8 women, how many ways can 2 men and 3 women be chosen for 5 positions

From a group of 10 men and 8 women, how many ways can 2 men and 3 women be chosen for 5 positions?
We use combinations. Since men and women are independent, we multiply each result:
We want 10 men choose 2 men multiplied by 8 women choose 3 women.
[URL='https://www.mathcelebrity.com/permutation.php?num=10&den=2&pl=Combinations']Type 10C2 into our search engine[/URL] and we get 45
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']Type 8C3 into our search engine[/URL] and we get 56
Multiply both together:
45 * 56 = [B]2,520 ways[/B]

From a standard 52 card deck, how many 6-card hands will have 2 spades and 4 hearts?

From a standard 52 card deck, how many 6-card hands will have 2 spades and 4 hearts?
We want the product of 13C2 * 13C4 since we have 13 possible spades choose 2 and 13 possible hearts choose 4
[LIST]
[*]Spades: 13C2 from our [URL='http://www.mathcelebrity.com/permutation.php?num=13&den=2&pl=Combinations']combinations calculator[/URL] = 78
[*]Hearts: 13C4 from our [URL='http://www.mathcelebrity.com/permutation.php?num=13&den=4&pl=Combinations']combinations calculator[/URL] = 715
[/LIST]
(78)(715) = [B]55,770[/B]

Group Combinations

Free Group Combinations Calculator - Given an original group of certain types of member, this determines how many groups/teams can be formed using a certain condition.

How many 4 person committees can be formed from a group of 11 people

How many 4 person committees can be formed from a group of 11 people
We want 11 choose 4, or 11C4.
We type [URL='https://www.mathcelebrity.com/permutation.php?num=11&den=4&pl=Combinations']11C4 into our search engine[/URL] and we get:
[B]330 committees[/B]

How many distinct 3 letter arrangements can be made using P, R, I, M and E

How many distinct 3 letter arrangements can be made using P, R, I, M and E?
We have all unique letters. We want the combination formula 5 Choose 3, or C(5,3).
Using our [URL='http://www.mathcelebrity.com/permutation.php?num=5&den=3&pl=Combinations']combinations calculator[/URL], we get 10 unique 3 letter arrangements.

How many ways can 4 trucks and 5 cars be selected from 15 cars and 12 trucks for a safety study

How many ways can 4 trucks and 5 cars be selected from 15 cars and 12 trucks for a safety study?
We have Truck combinations * Car combinations:
[URL='http://www.mathcelebrity.com/permutation.php?num=12&den=4&pl=Combinations']12C4[/URL] * [URL='http://www.mathcelebrity.com/permutation.php?num=15&den=5&pl=Combinations']15C5[/URL]
495 * 3,003
[B]1,486,485[/B]

How many ways can a basketball coach choose to the first five player from a group of 15 players

How many ways can a basketball coach choose to the first five player from a group of 15 players
We use combinations. We want 15 choose 5.
We type this in our search engine and we get:
[URL='https://www.mathcelebrity.com/permutation.php?num=15&den=5&pl=Combinations']15C5[/URL] = [B]3,003 different five player rosters[/B]

If 100 people are required to introduce themselves to each other and shake hands with each person on

If 100 people are required to introduce themselves to each other and shake hands with each person one time, how many handshakes will take place?
We want 100 choose 2 since we have 2 people per handshake:
[URL='https://www.mathcelebrity.com/permutation.php?num=100&den=2&pl=Combinations']100C2[/URL] = [B]4950[/B]

If there are 8 girls entered in a race, how many different ways can the runners place first, second,

If there are 8 girls entered in a race, how many different ways can the runners place first, second, and third?
We want 8 choose 3, or 8C3.
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']Type 8C3 into the search engine[/URL], and we get [B]56[/B] different ways to place first, second, and third.

In how many ways can a committee consisting of 4 faculty members and 5 students be formed if there a

In how many ways can a committee consisting of 4 faculty members and 5 students be formed if there are 8 faculty members and 9 students eligible to serve on the committee?
We have 8 choose 4 * 9 choose 5 written as : 8C4 * 9C5
[LIST]
[*][URL='http://www.mathcelebrity.com/permutation.php?num=8&den=4&pl=Combinations']8C4[/URL] = 70
[*][URL='http://www.mathcelebrity.com/permutation.php?num=9&den=5&pl=Combinations']9C5[/URL] = 126
[/LIST]
Multiply these together to get [B]8,820[/B]

In Maricopa County, 5 persons are to be elected to the Board of Supervisors. If 8 persons are candid

In Maricopa County, 5 persons are to be elected to the Board of Supervisors. If 8 persons are candidates, how many different arrangements are possible?
We want 8 choose 5, or 8C5. [URL='http://www.mathcelebrity.com/permutation.php?num=8&den=5&pl=Combinations']Typing this into the search engine[/URL] we get [B]56[/B].

License plate that is made up of 4 letters followed by 2 numbers

License plate that is made up of 4 letters followed by 2 numbers
Using the fundamental rule of counting, we have:
26 possible letters * 26 possible letters * 26 possible letters * 26 possible letters * 10 possible numbers * 10 possible numbers = [B]45,697,600 license plate combinations[/B]

license plate with 4 letter combinations and 3 number combinations

license plate with 4 letter combinations and 3 number combinations
There are 26 total letters and 10 digits [0-9].
We have 26 C 4 * 10 C 3.
[URL='http://www.mathcelebrity.com/permutation.php?num=26&den=4&pl=Combinations']26 C 4[/URL] = 14,950
[URL='http://www.mathcelebrity.com/permutation.php?num=10&den=3&pl=Combinations']10 C 3[/URL] = 120
Total license plate combinations:
14,950 * 120 = [B]1,794,000[/B]

Meryl can only take 4 out of 7 classes offered during the summer. How many different ways can she ch

Meryl can only take 4 out of 7 classes offered during the summer. How many different ways can she choose the classes she will take
We want 7 choose 4, or 7C4:
We [URL='https://www.mathcelebrity.com/permutation.php?num=7&den=4&pl=Combinations']type 7C4 into our search engine and we get[/URL]:
35

Mrs. Lopez gave a homework assignment over summer vacation to read three books from the following li

Mrs. Lopez gave a homework assignment over summer vacation to read three books from the following list:
a) Call of the Wild
b) Wuthering Heights
c) Death of a Salesman
d) The Cartoon Book of Physics
How many possible combinations of three books are there in the list of four books?
We need to elimination those of the same order, so we use combinations:
[URL='https://www.mathcelebrity.com/permutation.php?num=4&den=3&pl=Combinations']4C3[/URL] = [B]4[/B]

One thousand people in. room decide to shake hands with every other person in the room. Instead of o

One thousand people in. room decide to shake hands with every other person in the room. Instead of one handshake per couple, each person must shake both of the hands of every person in the room with both his right and his left hand. (Tom will use his right hand to shake Dave's right hand and then Dave's left hand. Tom will then use his left hand to shake Dave's right hand and then Dave's left hand.) How many total handshakes will take place?
1000 people taken 2 at a time:
[URL='https://www.mathcelebrity.com/permutation.php?num=1000&den=2&pl=Combinations']1000C2[/URL] = 499,500
But each group of 2 makes 4 unique handshakes:
499,500 * 4 = [B]1,998,000[/B]

Permutations and Combinations

Free Permutations and Combinations Calculator - Calculates the following:

Number of permutation(s) of n items arranged in r ways =_{n}P_{r}

Number of combination(s) of n items arranged in r__unique__ ways = _{n}C_{r} including subsets of sets

Number of permutation(s) of n items arranged in r ways =

Number of combination(s) of n items arranged in r

Select 6 bills from a combination of 5 different bills

We use the combination formula, 6 choose 5, or 6C5.
Using our [URL='http://www.mathcelebrity.com/permutation.php?num=6&den=5&pl=Combinations']combinations calculator[/URL], or entering 6C6 into the search engine, we get
[B]6 ways to select.[/B]

Serial numbers for a product are to made using 4 letters followed by 4 numbers. If the letters are t

Serial numbers for a product are to made using 4 letters followed by 4 numbers. If the letters are to be taken from the first 5 letters of the alphabet with repeats possible and the numbers are taken from the digits 0 through 9 with no repeats, how many serial numbers can be generated?
First 5 letters of the alphabet are {A, B, C, D, E}
The 4 letters can be chosen as possible:
5 * 5 * 5 * 5
The number are not repeatable, so the 4 numbers can be chosen as:
10 * 9 * 8 * 7 since we have one less choice with each pick
Grouping letters and numbers together, we have the following serial number combinations:
5 * 5 * 5 * 5 * 10 * 9 * 8 * 7 = [B]3,150,000[/B]

Seth is constantly forgetting the combination to his lock. He has a lock with four dials. (Each ha 1

Seth is constantly forgetting the combination to his lock. He has a lock with four dials. (Each has 10 numbers 0-9). If Seth can try one lock combination per second, how many seconds will it take him to try every possible lock combination?
Start with 0001, 0002, all the way to 9999
[URL='https://www.mathcelebrity.com/inclusnumwp.php?num1=0&num2=9999&pl=Count']When you do this[/URL], you get 10,000 combinations. One per second = 10,000 seconds

Suppose you are asked to choose three movies. If there are 20 different movies, in how many ways can

Suppose you are asked to choose three movies. If there are 20 different movies, in how many ways can the three movies be chosen?
We want unique combinations, so we have 20 choose 3:
We [URL='https://www.mathcelebrity.com/permutation.php?num=20&den=3&pl=Combinations']type 20C3 into our search engine[/URL] and we get:
20C3 = [B]1,140[/B]

Ten people are competing for the title of "Best Singer in the World". There will be a 1st place and

Ten people are competing for the title of "Best Singer in the World". There will be a 1st place and a 2nd place awarded. How many different ways can the 1st and 2nd place be awarded?:
We have a combinations problem of 10 choose 2.
Using our [URL='https://www.mathcelebrity.com/permutation.php?num=10&den=2&pl=Permutations']permutations calculator[/URL], we see that:
10 P 2 = [B]90 ways[/B]

The coach of a hockey team is holding tryouts and can take only 2 more players for the team. There a

The coach of a hockey team is holding tryouts and can take only 2 more players for the team. There are 5 players trying out. How many different groups of 2 players could possibly be chosen?
We want 5C2. Using our [URL='http://www.mathcelebrity.com/permutation.php?num=5&den=2&pl=Combinations']combinations calculator[/URL], we get:
[B]10[/B] possible combinations of 2 player groups

There are 24 competitors in a cycling race. How many different selections are possible for first and

There are 24 competitors in a cycling race. How many different selections are possible for first and second place?
We want unique combinations, so we have:
[URL='https://www.mathcelebrity.com/permutation.php?num=24&den=2&pl=Combinations']24 C 2[/URL] = [B]276[/B]

There are 6 women and 5 men in a department. How many ways can a committee of 2 women and 2 men be s

There are 6 women and 5 men in a department. How many ways can a committee of 2 women and 2 men be selected?
We want 6C2 * 5C2 using combinations.
[LIST]
[*]Using our [URL='http://www.mathcelebrity.com/permutation.php?num=6&den=2&pl=Combinations']combinations calculator[/URL], 6C2 = 15
[*]Using our [URL='http://www.mathcelebrity.com/permutation.php?num=5&den=2&pl=Combinations']combinations calculator[/URL], 5C2 = 10
[/LIST]
Our answer is 15 * 10 = [B]150[/B]

To create an entry code, you must first choose 2 letters and then, 4 single-digit numbers. How ma

To create an entry code, you must first choose 2 letters and then, 4 single-digit numbers. How many different entry codes can you create?
List total combinations using the product of all possibilities:
26 letters * 26 letters * 10 digits (0-9) * 10 digits (0-9) * 10 digits (0-9) * 10 digits (0-9)
[B]6,760,000[/B]

Tony has 6 cds that he is giving away. He lets his best friend choose 3 of the 6 cds. How many gr

Tony has 6 cds that he is giving away. He lets his best friend choose 3 of the 6 cds. How many groups of 3 cds are possible?
This problem asks for [I]unique[/I] combinations. We want 6 choose 3, or 6C3.
Go to the [URL='https://www.mathcelebrity.com/permutation.php?num=6&den=3&pl=Combinations']search engine, and type in 6C3[/URL], we get [B]20[/B] possible groups.

Twelve students tried out for the football team. Coach Moorhead only has 5 openings. In how many way

Twelve students tried out for the football team. Coach Moorhead only has 5 openings. In how many ways, can he pick 5 of the 12 students to be on the team?
We use the combinations formula. We can write this as 12C5. [URL='https://www.mathcelebrity.com/permutation.php?num=12&den=5&pl=Combinations']Type this into our search engine[/URL] and we get:
[B]792 ways[/B]

Vanessa is packing her bags for her vacation. She has 5 unique action figures, but only 3 fit in her

Vanessa is packing her bags for her vacation. She has 5 unique action figures, but only 3 fit in her bag. How many different groups of 3 action figures can she take?
The key word here is [U]different[/U]. This means combinations.
We use our [URL='http://www.mathcelebrity.com/permutation.php?num=5&den=3&pl=Combinations']combinations calculator[/URL] to find 5 C 3 which equals [B]10[/B].