Solution: 2018 Winter Final - 1

Author: Michiel Smid

Question

You are given 20 beer bottles $B_1,B_2,\dots,B_{20}$ and 50 cider bottles $C_1,C_2,\dots,C_{50}$. Consider subsets of these 70 bottles, consisting of 30 bottles, exactly 12 of which are beer bottles. How many subsets are there?

(a)

${50 \choose 12} \cdot {20 \choose 18}$

(b)

${20 \choose 12} \cdot {38 \choose 18}$

(c)

${70 \choose 30} \cdot {20 \choose 12}$

(d)

${20 \choose 12} \cdot {50 \choose 18}$

Solution

Let’s choose 12 beer bottles out of the 20: $ \binom{20}{12} $

Let’s choose 18 cider bottles out of the 50: $ \binom{50}{18} $

$ \binom{20}{12} \cdot \binom{50}{18} $