N2011 P2 Q11

A committee of 10 people is chosen at random from a group consisting of 18 women and 12 men. The number of women on the committee is denoted by \(R\).

1. Find the probability that \(R=4\).[3]
2. The most probable number of women on the committee is denoted by \(r\). By using the fact that \(\mathrm P(R=r)>\mathrm P(R=r+1)\), show that \(r\) satisfies the inequality \((r+1)!(17-r)!(9-r)!(r+3)!>r!(18-r)!(10-r)!(r+2)!\), and use this inequality to find the value of \(r\).[5]