ten students from a class volunteer to be on a committe. how many ways can 6 be chosen for the committee ?

Question

please help me with that question, and how do you know if that is a permutation or a combination?
besides having order matter in a permutation and in a combination order not matter, what is another thing that is significant between them? i just cant tell the difference!

thanks in advance :D

Answer 1

Permutation and combination are often very confusing. Even today I have to think about which word means what. Sometimes you even hear people say a combination when they mean a permutation or vice versa.

Permutation is an *arrangement* of the people where order matters. (Permutation comes from the Latin for "rearrangement" so you should always think *order*)

Combination is a selection of the people *without* respect to order. It's just how you are going to "combine" them into a set, but Amy and Bob is the same as Bob and Amy.

You are picking 6 people to be on the committee, but you don't care whether you pick Bob first or second or third. You just care if he is on the committee. So this is a *combination* problem.

Next. How to solve?

You have 10 choices for the first person.
9 choices for the second person
8 choices for the third person
7 choices for the fourth person
6 choices for the fifth person
5 choices for the fourth person

All together you have 10 x 9 x 8 x 7 x 6 x 5 ways to pick the committee. However, in picking these 6 people, order doesn't matter, so you need to divide by 6! so you don't over count.

If I've calculated it correctly that would be 210 ways to pick a six person committee from a class of ten people.

The formula for combinations is:
nCk = n! / k! (n-k)!

The formula for permutations is:
nPk = n! / (n-k)!

Answer 2

Order is the only difference between a permutation and a combination. Both are collections, but a permutation has order while a combination doesn't. That's it.

Clearly for collections of the same size, there will be fewer combinations than permutations.

From the wording of the question, it's safe to assume that the order in which the students volunteer doesn't matter. So we are looking at a combination, not a permutation.

So, using the combination formula, there are 10! / (4! 6!) ways of choosing the committee.