Question: 2018 Fall Final - 20

Author: Michiel Smid

Let $n \geq 2$ be an integer. You are given $n$ beer bottles $B_1,B_2,\dots,B_n$ and two cider bottles $C_1$ and $C_2$. You choose a uniformly random 3-element subset of the set of these $n+2$ bottles. Define the random variable $X$ to be

X = the number of cider bottles in the chosen subset.

What is the expected value $\mathbb{E}(X)$ of the random variable $X$?

(a)

$\frac{2 {{n}\choose{2}} + n - 1}{{{n+2}\choose{3}}}$

(b)

$\frac{2 {{n}\choose{2}} + n + 1}{{{n+2}\choose{3}}}$

(c)

$\frac{2 {{n}\choose{2}} + 2n}{{{n+2}\choose{3}}}$

(d)

$\frac{2 {{n}\choose{2}} + n}{{{n+2}\choose{3}}}$