1982 TYS

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success in any one trial is $p$.

Show that

$\frac{\text{P}\left( X=k+1 \right)}{\text{P}\left( X=k \right)}=\frac{\left( n-k \right)p}{\left( k+1 \right)\left( 1-p \right)},k=0,1,2,...,\left( n-1 \right)$

Find the most probable number of success when $n=10$ and $p=\frac{1}{4}$.