Question

Why is `x= _3C_9` impossible to evaluate?

Answer 1

It's not impossible to evaluate: it's just 0.

The best way to think of `\ _nC_r` is as "n choose r", or "how many ways can I choose r things from n things?"

In your case, that would mean "how many ways can I choose 9 things from 3 things?" If I only have 3 things, there is no way I can choose 9 things. Hence, there are 0 possible ways to do it.

If you wanted to consider `\ _9C_3`, we can easily calculate that:
`\ _9C_3 = (9!)/(3!6!) = (9*8*7)/(3*2*1) = 3 * 4 * 7 = 84`