What is n if nC5=nC3?

Answer 1

8, because choosing 5 items to put in a group, is the same as finding the ways to leave 3 items out of the group.

You can imagine that you had 8 people and you want to find the ways to choose 5 of them to play baseball. That's 8 choose 5 (56 ways). But for each team of 5 you select, there is a set of 3 people that won't be playing. Obviously the numbers should be the same. Indeed 8 choose 3 (number of ways to pick people to sit out the game) is also 56 ways.

In general,
nCk = n! / k! (n-k)!

In your case you want:
n! / 5! (n-5)! = n! / 3! (n-3)!

The numerators are the same, so you have to equate the denominators:
5! (n-5)! = 3! (n-3)!

You can calculate it all out, but the obvious answer is 8 because then you get 5!3! = 3!5!

Now that you know the secret, tell me the value of n if nC6 = nC2?