permutation - a way in which a set or number of things can be ordered or arranged.

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

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 baseball team has 25 total players consisting of 15 position players and 10 pitchers. How many dif

A baseball team has 25 total players consisting of 15 position players and 10 pitchers. How many different ways are there to arrange the batting order of 9 starting players if only one pitcher is used at a time and the pitcher always bats last.
(This means that 8 players are taken from the position players and one pitcher is then added at the end of the lineup.)
First 8 positions:
[URL='https://www.mathcelebrity.com/permutation.php?num=15&den=8&pl=Permutations']15P8[/URL] = 259,459,200
For the pitcher, we can have 10 different possibilities for the 9th player:
259,459,200 x 10 = [B]2,594,592,000 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 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 mathematician has 8 favorite paintings and only 6 wall hooks to hang the paintings. How many diffe

A mathematician has 8 favorite paintings and only 6 wall hooks to hang the paintings. How many different ways can she hang the paintings?
8 paintings taken 6 at a time is written as:
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=6&pl=Permutations']8P6[/URL] = [B]20,160[/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 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]

An eccentric millionaire has 5 golden hooks from which to hang her expensive artwork. She wants to h

An eccentric millionaire has 5 golden hooks from which to hang her expensive artwork. She wants to have enough paintings so she can change the order of the arrangement each day for the next 41 years. (The same five paintings are okay as long as the hanging order is different.) What is the fewest number of paintings she can buy and still have a different arrangement every day for the next 41 years?
365 days * 41 years + 10 leap year days = 14,975 days
what is the lowest permutations count of n such that nP5 >= 14,975
W[URL='https://www.mathcelebrity.com/permutation.php?num=9&den=5&pl=Permutations']e see that 9P5[/URL] = 15,120, so the answer is [B]9 paintings[/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].

Circular Permutation

Free Circular Permutation Calculator - Calculates the following:

Number of ways to arrange n distinct items arranged on a circle

Number of ways to arrange n distinct items arranged on a circle

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 r in P(7, r)

Find r in P(7, r)
Recall the permutations formula:
7! / (7-r!) = 840.
We [URL='https://www.mathcelebrity.com/factorial.php?num=7!&pl=Calculate+factorial']run 7! through our search engine[/URL] and we get:
[URL='https://www.mathcelebrity.com/factorial.php?num=7!&pl=Calculate+factorial']7![/URL] = 5040
5040 / (7 - r)! = 840
Cross multiply, and we get:
5040/840 = 7 - r!
6 = (7 - r)!
Since 6 = 3*2*! = 3!, we have;
3! = (7 - r)!
3 = 7 - r
To solve for r, we [URL='https://www.mathcelebrity.com/1unk.php?num=3%3D7-r&pl=Solve']type this equation into our search engine[/URL] and we get:
r = [B]4[/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]

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 different ways could you arrange 5 books on a shelf

How many different ways could you arrange 5 books on a shelf?
[URL='https://www.mathcelebrity.com/factorial.php?num=5!&pl=Calculate+factorial']Using permutations, you can type in 5![/URL] and we get:
[B]120 different ways.[/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 5 people be seated in 5 seats?

How many ways can 5 people be seated in 5 seats?
We have the permutation 5!.
Because the first seat can have 5 different people.
The next seat has 5 - 1 = 4 people since one person is in the first seat
The next seat can have 5 - 2 = 3 people since we have two people in the first two seats
The next seat can have 5 - 3 = 2 people since we have three people in the first three seats
The next seat can have 5 - 4 = 1 people since we have four people in the first four seats
[URL='https://www.mathcelebrity.com/factorial.php?num=5!&pl=Calculate+factorial']Type in 5! into our search engine[/URL], and we get 120.

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].

Jay has 5 paintings that he plans to display on a wall that only has 4 books. Nancy has 5 paintings

Jay has 5 paintings that he plans to display on a wall that only has 4 books. Nancy has 5 paintings that she plans to display on a wall with 5 hooks. Who has more possible ways to hang his/her paintings?
Jay's ways:
[URL='https://www.mathcelebrity.com/permutation.php?num=5&den=4&pl=Permutations']5 P 4 [/URL]= [B]120
[/B]
Nancy's ways:
[URL='https://www.mathcelebrity.com/permutation.php?num=5&den=5&pl=Permutations']5 P 5[/URL] = [B]120
Therefore, they have the same number of ways.[/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

Permutations with Replacement

Free Permutations with Replacement Calculator - Calculates the following:

How many permutations 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 permutations 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?

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]

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]

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 friends went to a movie theater. Because the movie was boring, they decided to figure out how

Twelve friends went to a movie theater. Because the movie was boring, they decided to figure out how many different ways they could sit in the 12 seats. How many different permutations are there for these 12 friends?
12 taken 12 at a time is written as:
[URL='https://www.mathcelebrity.com/permutation.php?num=12&den=12&pl=Permutations']12P12[/URL] = [B]479,001,600[/B]

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].