Permutations and Combinations

In this lesson we single out two important special cases of the Fundamental Counting Principle—permutations and combinations.

Goal: Identity when to use permutations and combinations.

Permutation

A permutation is one of the different arrangements of a group of items where order matters.

A permutation of a set of distinct objects is an ordering of these objects.

Anytime you see “order”, plug your numbers into the permutation equation.

Permutations give really big numbers!!

Permutation Example

Some permutations of the letters ABCDWXYZ are

XAYBZWCD ZAYBCDWX DBWAZXYC

YDXAWCZB How many such permutations are

possible? 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

Now, there is a short cut to writing out 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

Its called a factorial, and it looks like an exclamation mark (!).

The number of permutations of n objects is n!.

8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

4! = 4 x 3 x 2 x 1 = 24

How many permutations consisting of five letters can be made from these same eight letters? (ABCDWXYZ)

Some are: XYZWC AZDWX AZXYB WDXZB

By the Fundamental Counting Principle, the number of such permutations is

8 x 7 x 6 x 5 x 4 = 6720But there is another shortcut….

Permutation Formula

If a set has n elements, then the number of ways of ordering r elements from the set is denoted by P(n, r).

(n = the number of elements you can choose from; r = how many you are actually going to use)

So, from the question of: How many permutations consisting of five letters can be made from these same eight letters? (ABCDWXYZ)

P(8,5)

Lets try this with students…

Example 1:

A club has nine members. In how many ways can a president, vice president, and secretary be chosen from the members of this club?

Does order matter? Yes, then it is a permutation, we can

use the permutation formula. P(9, 3) =

= 504

Example 2: From 20 raffle tickets in a hat, four

tickets are to be selected in order. The holder of the first ticket wins a car, the second a motorcycle, the third a bicycle, and the fourth a skateboard. In how many different ways can these prizes be awarded?

Does order matter? Yes, then it is a permutation, we can use

the permutation formula. P(20, 4) =

Raffle tickets cont….

P(20,4) =

116,280

Now lets practice….

Combinations

When finding permutations, we are interested in the number of ways of ordering elements of a set. In many counting problems, however, order is not important…

Combinations

A combination of r elements of a set is any subset of r elements from the set (Order does not matter).

If the set has n elements, then the number of combinations of r elements is denoted by C(n, r).

Combinations give smaller numbers!!

Combination Formula

• The key difference between permutations and combinations is order. If we are interested in ordered arrangements, then we are counting permutations; but if we are concerned with subsets without regard to order, then we are counting combinations.

How to tell if order matters… Which one is a permutation and which is a combination?

A coach must choose five starters from a team of 12 players. How many different ways can the coach choose the starters?

How many different ways can the coach select the 1st star, 2nd star, and 3rd star of the game?

Example 1:

A club has nine members. In how many ways can a committee of three be chosen from the members of this club?

Does order matter? No, then it is a combination. C(9,3)=

= 84

Example 2:

From 20 raffle tickets in a hat, four tickets are to be chosen at random. The holders of the winning tickets are to be awarded free trips to the Bahamas. In how many ways can the four winners be chosen?

C(20,4)=

= 4845

Example 3:

There are fourteen juniors and three seniors in the Service Club. The club is to send four representatives to the State Conference. How many different ways are there to select a group of four students to attend the conference?

C(17,4)= 2380

Now practice…..