Discrete Random Variables (DRV)

A bag containing \((n+5)\) numbered balls. Two of the balls are numbered \(3\), three of the balls are numbered \(4\) and \(n\) of the balls are numbered \(5\). Two balls are taken, at random and without replacement, from the bag. The random variable \(S\) is the sum of the numbers on the two balls taken.

1. Determine the probability distribution of \(S\).[4]
2. For the case where \(n=1\), find \(P(S=10)\) and explain this result.[1]
3. Show that \(E(S)=\frac{10n+36}{n+5}\) and \(\text{Var}\hspace{0.1em}(S)=\frac{g(n)}{(n+5)^{2}(n+4)}\) where \(g(n)\) is a quadratic polynomial to be determined.[6]