Factorials and Their Applications

1. Definition of the Factorial

We define n! recursively by

0! = 0,    1! = 1,    n! = n(n - 1)!

Example:

5! = 5(4)(3)(2) = 120

Example:

Suppose that we are interested in how many ways there are in scrambling the letters of the name "Cindy".  We have 5 choices for the first letter, once we have chosen the first letter there are 4 choices for the second letter, and then three choices for the third letter, two for the fourth letter, and only one choice for the last letter.  Hence there are

5(4)(3)(2)(1) = 5!

choices.

2. Permutations

Example

If we want to select only three letters from the word "Cindy" then we have

(5)(4)(3) = 5!/(5 - 3)!

choices.

 Definition The number of permutations of n distinct objects taken r at a time is           nPr = n!/(n - r)!

You can find this button on the TI 85 calculator by hitting Math -> Prob

3. Distinguishable Permutations

Example

How many ways are there of scrambling the name Tamara Heether?

Solution:

If there were no duplicate letters the solution would be 13!, but this is not the case.  There are

2 T's,     3 A's     2 R's
and     3E's

We must divide by 2!3!2!3! to get

13!/[2! 3! 2! 3!] = 43,243,200

 Theorem If there are n objects with n1 duplicates of one kind, n2 duplicates of a second kind, ..., nk duplicates of a kth kind, then the number of distinguishable permutations of these n objects is           n!/(n1!n2!...nk!)

Exercise:

How many ways are there to scramble your first and last name?

4. Combinations

Example

How many different five card poker hands are there?

Solution

First note that there are 52P5 different ordered five card poker hands, however, two hands that have the same five cards, but in a different order should not be counted as distinct hands.  Since there are 5! ways of ordering five cards, we have

52P5/5! = 52!/[5!(52 - 5)!] = 2,598,960

different poker hands.

Note that only four of these hands are Royal Flushes, hence there is a 4 in 2,598,960 or about one in half a million chance of receiving a Royal Flush in a 5 card stud poker game.

 Theorem  The number of ways of choosing r objects from n where order does not matter is           nCr = n!/(n - r)!r!

5. The Binomial Theorem

consider

(x + y)5  = (x + y)(x + y)(x + y)(x + y)(x + y)

Q:  How many ways are there to select all x's?

A:  1 way.

Q:  How many ways are there to select 4x's from the 5 possible?

A:  5C4    ways

Exercise:

How many ways are there to select two x's from the five?

These investigations lead us to believe that

(x + y)5  = 5C5 x5  + 5C4 x4y + 5C3 x3y2 + 5C2 x2y3 + 5C1 xy4 5C0 y5

 Theorem           (x + y)n = Si = 0n nCn - i xn-iyi

Example

Find

(3x - 2y)4

Solution

The formula gives us

5C4 (3x)4 + 5C3 (3x)3(-2y) + 5C2 (3 x)2(-2y)2 + 5C1 (3 x)(-2y)3 5C0 (-2y)4

=  5(34x4) + 10(27x3)(-2y) + 10(9x2)(4y2) + 5(3x)(-8y3) + (16y4)

= 405x4 - 540x3y + 360x2y2 - 120xy3 + 16y4