[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Show that

[[math]] b(n,p,j) = \frac pq \left(\frac {n - j + 1}j \right) b(n,p,j - 1)\ , [[/math]]

for [math]j \ge 1[/math]. Use this fact to determine the value or values of [math]j[/math] which give [math]b(n,p,j)[/math] its greatest value. Hint: Consider the successive ratios as [math]j[/math] increases.