Solution: 2016 Fall Final - 13

Author: Michiel Smid

Question

Let $V$ be a set consisting of 12 even integers and 8 odd integers. We choose a uniformly random subset $W$ of $V$ having size 7. Define the event

What is $\Pr(A)$?

(a)

$\frac{{20 \choose 3}{17 \choose 4}}{20 \choose 7}$

(b)

$\frac{{12 \choose 4}{8 \choose 3}}{20 \choose 7}$

(c)

$\frac{{20 \choose 4}{16 \choose 3}}{20 \choose 7}$

(d)

$\frac{{12 \choose 4}+{8 \choose 3}}{20 \choose 7}$

Let's determine $ |V| $
$ |V| = 20 $
Let's determine $ |W| $
$ |W| = 7 $
Let's determine $ |A| $
The number of ways to choose 4 even integers from 12 even integers: $ \binom{12}{4} $
The number of ways to choose 3 odd integers from 8 odd integers: $ \binom{8}{3} $
$ |A| = \binom{12}{4} \binom{8}{3} $
Let's determine $ Pr(A) $
$ Pr(A) = \frac{\binom{12}{4} \binom{8}{3}}{\binom{20}{7}} $