factorial - The product of an integer and all the integers below it

A mother duck lines her 13 ducklings up behind her. In how many ways can the ducklings line up

A mother duck lines her 13 ducklings up behind her. In how many ways can the ducklings line up?
In position one, we can have any of the 13 ducks.
In position two, we can have 12 ducks, since one has to occupy position one.
We subtract 1 each time until we fill up all 13 positions.
We have:
13 * 12 * 11 * ... * 2 * 1
Or, 13!. [URL='https://www.mathcelebrity.com/factorial.php?num=13!&pl=Calculate+factorial']Typing 13! into our search engine[/URL], we get [B]6,227,020,800[/B] ways the ducklings can line up behind the mother duck.

A tourist in Ireland wants to visit six different cities. How many different routes are possible?

A tourist in Ireland wants to visit six different cities. How many different routes are possible?
[URL='https://www.mathcelebrity.com/factorial.php?num=6!&pl=Calculate+factorial']We want 6![/URL] which is [B]720[/B]

Derangements - Subfactorials

Free Derangements - Subfactorials Calculator - Calculates the number of derangements/subfactorial !n.

Factorials

Free Factorials Calculator - Calculates the following factorial items:

* A factorial of one number such as n!

* A factorial of a numerator divided by a factorial of a denominator such as n!m!/a!b!

* Double Factorials such as n!!

* Stirlings Approximation for n!

* A factorial of one number such as n!

* A factorial of a numerator divided by a factorial of a denominator such as n!m!/a!b!

* Double Factorials such as n!!

* Stirlings Approximation for n!

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]

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 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.

Make 19 using only four four's

Make 19 using only four four's
4!-4-4/4
To prove our work, we have:
[URL='https://www.mathcelebrity.com/factorial.php?num=4!&pl=Calculate+factorial']4![/URL] = 24
24 - 4 = 20
Since 4/4 = 1, we have:
20 - 1 = 19

Multifactorials

Free Multifactorials Calculator - Calculates the multifactorial n!^{(m)}

Prove 0! = 1

Prove 0! = 1
Let n be a whole number, where n! represents the product of n and all integers below it through 1.
The factorial formula for n is:
n! = n · (n - 1) * (n - 2) * ... * 3 * 2 * 1
Written in partially expanded form, n! is:
n! = n * (n - 1)!
[U]Substitute n = 1 into this expression:[/U]
n! = n * (n - 1)!
1! = 1 * (1 - 1)!
1! = 1 * (0)!
For the expression to be true, 0! [U]must[/U] equal 1. Otherwise, 1! <> 1 which contradicts the equation above

The coach writes the batting order on a piece of paper. How many different ways could the list be wr

The coach writes the batting order on a piece of paper. How many different ways could the list be written?
We have 9 people in a line up. The total lineups are shown by:
9 * 8 * 7 * ... * 2 * 1
Or, 9!. [URL='https://www.mathcelebrity.com/factorial.php?num=9!&pl=Calculate+factorial']Typing 9! in our search engine[/URL] and we get [B]362,880[/B]

Write in set builder form {all possible numbers formed by any two of the digits 1 2 5}

Write in set builder form {all possible numbers formed by any two of the digits 1 2 5}
With 3 numbers, we got [URL='https://www.mathcelebrity.com/factorial.php?num=3!&pl=Calculate+factorial']3! = 6[/URL] possible numbers formed by the two digits
[LIST=1]
[*]12
[*]15
[*]21
[*]25
[*]51
[*]52
[/LIST]
In set builder notation, we write this as:
{x : x ? {12, 15, 21, 25, 51, 52})
x such that x is a element of the set {12, 15, 21, 25, 51, 52}

you and 5 friends go to a concert. how many different ways can you sit in the assigned seats

You and 5 friends go to a concert. how many different ways can you sit in the assigned seats?
With 6 possible seats, the [URL='https://www.mathcelebrity.com/factorial.php?num=6!&pl=Calculate+factorial']number of unique arrangements is[/URL]:
6! = 6 x 5 x 4 x 3 x 2 x 1 = [B]720[/B]