Chapter 15.9, Problem 10P

Derive equation ( 9.9 ) as follows: In $C(n,x)$, show that $n!/(n-x)!~n^x$ for fixed $x$ and large $n$ [write $n!/(n-x)!$ as a product of $x$ factors, divide by $n^x$, and show that the limit is 1 as $n\to \infty$ ]. Then write $q^{n-x}={(1-p)}^{n-x}$ as ${(1-p)}^n{(1-p)}^{-x}=$ ${(1-np/n)}^n{(1-p)}^{-x};$ evaluate the limit of the first factor as $n\to \infty ,np$ fixed; the limit of the second factor as $p\to 0$ is 1. Collect your results to obtain equation (9.9).