Question

For an even number of objects(say n), the number of combinations of n things taken r at a time is greatest when r=n/2. Why is this so? I am searching for a logical, explanatory answer. Mathematical proofs are difficult to remember.

Take the question this way. Suppose you have to select a new team of moderators from the 8 best members of openstudy. Why should the no of choices be greatest when you choose to select only 4 out of 8 members?
Is my question clear? If not, please tell me

Answer 1

Combinations:
nCk = n! / [ (n - k)!*k! ]

If k is small, then (n - k)! is a much bigger number. Since it's in the denominator, the entire thing is smaller.

If k is large, k! gets really big and, by the same argument, makes the entire thing smaller.

k = n/2 is a nice compromise. Not too small to make (n - k)! blow up and not too large to make k! affect things.

Did that make sense?

Answer 2

Well, very good. I never thought that someone would reply to his. Thanks!