binomial coefficient where k > n

Question

For solving binomial coefficients we have use from formula

$\frac{n!}{k!(n-k)!}$

This formula only works if n > k.

What happens if n < k? Is there another formula we need to use?

Answer 1

Your definition of "binomial coefficient" is too narrow. If $k$ is a nonnegative integer and $n$ is any real or complex number, the binomial coefficient $\binom nk$ is defined as follows: $$\binom nk=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}.$$ This is how the coefficient is understood in the general binomial theorem: $$(1+x)^n=\sum_{k=0}^\infty\binom nkx^k.$$

The formula $\binom nk=\frac{n!}{k!(n-k)!}$ only applies in the special case where $n$ is an integer and $n\ge k.$

To answer your question, note that $\binom nk$ is a polynomial of degree $k$ in the variable $n,$ and its zeros are $0,1,2,\dots,k-1.$ Thus $\binom nk=0$ if $n$ is a nonnegative integer $\lt k.$ On the other hand, $\binom nk\ne0$ if $n$ is negative or fractional or non-real.