Combinations

An arrangement of objects in which the order is not important is called a combination. This is different from permutation where the order matters. For example, suppose we are arranging the letters A, B and C. In a permutation, the arrangement ABC and ACB are different. But, in a combination, the arrangements ABC and ACB are the same because the order is not important.

The number of combinations of n things taken r at a time is written as C(n, r).

The formula is given by:

If you are not familiar with the n! (n factorial notation) then have a look the factorial lesson.

Example:

In how many ways can a coach choose three swimmers from among five swimmers?

Solution:

There are 5 swimmers to be taken 3 at a time.

Using the formula:

The coach can choose the swimmers in 10 ways.

Example:

Six friends want to play enough games of chess to be sure every one plays everyone else. How many games will they have to play?

Solution:

There are 6 players to be taken 2 at a time.

Using the formula:

They will need to play 15 games.

Example:

In a lottery, each ticket has 5 one-digit numbers 0-9 on it.

a) You win if your ticket has the digits in any order. What are your changes of winning?

b) You would win only if your ticket has the digits in the required order. What are your chances of winning?

Solution:

There are 10 digits to be taken 5 at a time.

a) Using the formula:

The chances of winning are 1 out of 252.

b) Since the order matters, we should use permutation instead of combination.

P(10, 5) = 10 x 9 x 8 x 7 x 6 = 30240

The chances of winning are 1 out of 30240.

The following video provides some information on combinations and how to solve some word problems using combinations.

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