Consider the possible arrangements of the letters a, b, and c.  List the outcomes in the sample space. If the order is important, then

each arrangement is a permutation of the three letters

Outcomes in the sample space.

abc    acb    bac     bca    cab    cba

There are six permutations.

Permutation Formula:Use when the item possibilities are all

different.n: number of available items or choices

r: the number of items to be selected    Sometimes this formula is written: P(n,r).

• Remember: The permutation formula takes order into account. The outcome (red, blue, green) is not the same as the outcome (green, blue, red.)

If n=5 and r=2, what is the value of 5P2? The symbol 5P2 means the number of

permutations of 5 items taken 2 at a time. 

• A permutation of 5 objects taken 2 at a time is equal to . . .

If there are n items with n1 alike, n2 alike, n3 alike, . . ., nk alike, the number of

permutations is calculated by dividing n factorial by the product of the factorials of

the number of occurences of each of the like items.

Use this permutation formula when some of the item possibilities are alike.

How many permutations are there for the letters in the word, BANANA?

First, notice there are six total letters of which the A is repeated 3 times and the N

is repeated 2 times. The number of permutations is equal to . . . .  There are six letters in BANANA (6!).

The A repeats three times (3!)and the N repeats two times (2!).

There are 60 permutations.

How many permutations are there for the letters in the word, HAPPY?

Remember to ask yourself, how many total letters are there?  How many letters are repeated? How many times is each

letter repeated?   There are five letters in HAPPY

(5!).The P repeats two times (2!).

There are

1) Calculate the value of 7P3

A permutation of 7 objects taken 3 ata time is equal to . . .

2) Calculate the value of 9P4

A permutation of 9 objects taken 4 at a time is equal to . . .

How many permutations are there for the letters in the word,

STATISTICS?

There are ten letters in STATISTICS (10!).

The S repeats three times (3!),the T repeats three times (3!),

and the I repeats two times (2!).

When different orderings of the same items are to be counted individually, i.e.

each listing represents a different scenario (mn is not the same as nm), the problem

involves permutations.

When different orderings are not to be counted separately, i.e. the outcome, mn is equivalent to the outcome nm, the problem

involves combinations.

Combination Formula:Different orders of the same items are not

counted.  The combination formula is equivalent to dividing the corresponding

number of permutations by r!.n: number of available items or choices

r: the number of items to be selected    Sometimes this formula is written: C(n,r).

Taking the letters a, b, and c taken two at a time, there are six permutations:

{ab, ac, ba, bc, ca, cb}.  If the order of the arrangement is not important, how

many of these outcomes are equivalent, i.e. how many combinations are there

ab = ba; ac = ca; and bc = cbThe three duplicate permutations would not be counted, therefore

three combinations exist

Calculate the value of 7C4.

This represents a combination of 7 objects

taken 4 at a time and is equal to

Calculate the value of 9C5

This represents a combination of 9 objects taken 5 at a time

and is equal to . . .

Determine whether the following scenarios represent permutations or combinations. Correct answers are

provided at the end of this section.1) Selecting two types of yogurt from the grocery's

dairy case from a selection of nine.2) Selecting your favorite yogurt and then your second favorite yogurt from a selection of nine.

3) Selecting three members from your class to work specific homework problems on the board.

4) Choosing two books to take with you on vacation from the nine books on your shelf.

5) Choosing three CDs to purchase from the music store.

6) Arranging seven photographs on a page of your senior memory book.

• 1) Combination2) Permutation3) Permutation4) Combination5) Combination6) Permutation

In how many ways can three class representatives be chosen from a group of

twelve students?  If the order of the arrangement is not important, how many

outcomes will there be?  This represents a combination of 12 objects

taken 3 at a time and is equal to 9!

9!9!