How many four person committees can be chosen from a group of eight people?

Answer 1

Two four person committees can be made from a group of eight people.

Because 8/4=2.

Sorry, but that answer is incorrect, so here is the way to the correct answer.

Imagine that the 8 people are standing, and there are only four empty chairs available.

In how many different ways can you choose any one person to sit on the first chair ? (and the answer is 8).

Then in how many ways can you fill the second chair from the remaining 7 people?

(and the answer is 7).

Which means that for filling the first two chairs there are 8 x 7 = 56 ways.

Similarly to fill the first 3 chairs there are 8 x 7 x 6 = 336 ways.

And to fill the four chairs, the answer is 8 x 7 x 6 x 5 = 1680 ways.

No wonder committees argue so much!

That would be true if the order of the seating matters. However, consider the committee consisting of Alf, Bert, Charlotte and Diana. They could sit in the order DCAB. However, suppose they sat down in another order, for example CDBA or CADB. These committees would still consist of the same 4 members ABCD (and the discussions would still be the same).

Any of the 4 ABCD could be the first member of the committee to sit, any of the remaining 3 could sit next, and so on, making a total of 4 x 3 x 2 x 1 = 24 ways the committee would consist of ABCD (in some order).

This would be true of all the other 1656 possible committees - there will be groups of 24 committees which have the same members seated in a different order - so there would only be 1680 ÷ 24 = 70 different committees.