Is inverse of a combination function negligible?

Question

Can anyone help me in determining whether $1/{n\choose a}$ negligible function for sufficiently large value of $n$, say for example $n=p^2$ and $a=p$, for an integer $p$?

Answer 1

Hint:

$$1/{n\choose a}=\frac{a!(n-a)!}{n!}=\frac{p!(p^2-p)!}{(p^2)!}=\frac{p!(p^2)!}{(p^2)!\prod^{p-1}_{i=1}(p^2-i)}=\frac{p!}{\prod^{p-1}_{i=0}(p^2-i)}$$

Now write the nominator as a product series as well. Then adjust the indices and use multiplicative factors to shift indices. Then write it as one big product. Find an upper bound for the fraction inside the product. Replace the fraction, making the multiplied term independent of the variable. Conclude.