2018 ACJC MYE Q8

The random variable \(X\) is the number of successes in \(n\) independent trials of an experiment in which the probability of a success at any trial is \(p\).

Denoting \(\mathrm{P}(X=k)\) by \({{p}_{k}}\), it is given that \(\frac{{{p}_{k}}}{{{p}_{k-1}}}=\frac{(n-k+1)p}{k(1-p)}\), \(k=1\), \(2\), …, \(n\).

A distribution is said to be bimodal if it has two modes.

Find the least value of \(n\), and the corresponding modes of \(X\), given that \(X\) is bimodal and that \(p=\frac{18}{25}\).