Question

50 members on committee. break down into 4 person subcommitees. How many possibilities?

Answer 1

but 50 is not divisible by 4.

Answer 2

50*49*48*47=5527200

Answer 3

You're looking for the number given by\[50C4=\frac{50!}{4!(50-4)!}=\frac{50!}{4!46!}=\frac{50\cdot49\cdot48\cdot47}{4\cdot3\cdot2\cdot1}\]Simplify this out, we get \[\frac{5527200}{24}=230300\]Total possible subcommittees.

Answer 4

There are 50 choices for the first member, then 49 choices for the second, et cetera, for a total of 50*49*48*47 = 5,527,200. If it matters what order the people are chosen, then this is the answer. For example, if there is some kind of seniority, and the first one chosen is the chairman, the second vice-chairman, the third secretary and the fourth treasurer.

However, it is much more likely that you don't care about the order of choice. That is, the committee formed by choosing Carl, Gandalf, Gollum and Morgoth in that order is the same as choosing Gandalf, Gollum, Morgoth and Carl in that order. In that case, you have a massive overcounting, because you count committees formed in different order as different. Fortunately, it's easy to fix this. How many different orders of the people on the committee are there? 4*3*2 = 24. Hence you have listed 24 times more committees than there really are, because you counted each of those 24 possible ways of picking the committee separately. To fix the problem, divide the number by 24: 5,527,200 / 24 = 230,300.

That's a longer explanation, but it may suit you better than just memorizing the formula, because it relies on thinking the situation out logically.