Number of Ways to Make A Group

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Permutation vs Combination
Two examples are given on this page. Each shows how a permutation and a combination relate to the problem.

Permutations and combinations are short cuts to counting when the tree diagram is not required or too large.

A permutation is used to count the number of results of an experiment in which order counts, lists are made.

A combination is used to count the number of results of an experiment in which order doesn't count, groups are made.
Ann, Martin, Nancy, and Tom Form A Club of 4 Members
a.) They must choose a president, secretary, and treasurer.
b.) They must choose a committee with 3 members.
On the far left the tree diagram which generates the sample space of officers is listed. In the middle the 24 officers are listed. On the left the 4 possible committees are listed with duplicates connected.

The permutation for the number in an ordered list of officers is on the left. The combination for the number of unordered groups or committees is on the right.
A club has 4 members.

a.) In how many ways can 2 officers be chosen?
b.) In how many ways can a committee of 2 member be chosen?

There are 4 x 3 or 12 ways to choose the 2 officers.

To choose the committee of 2, order does not count, so, AB is really BA and should not be counted as different committee.

This time a combination is used: 12/2! or 12/2 or 6 ways. A combination is a permutation divided to remove duplicates.

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