Solution: 2018 Winter Final - 2

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, that contain exactly 12 beer bottles (and any number of cider bottles) or exactly 12 cider bottles (and any number of beer bottles). How many such subsets are there?

(a)

${20 \choose 12} + {50 \choose 12} - {20 \choose 12} \cdot {50 \choose 12}$

(b)

${20 \choose 12} \cdot 2^{50} + {50 \choose 12} \cdot 2^{20} - {20 \choose 12} \cdot {50 \choose 12}$

(c)

${20 \choose 12} + {50 \choose 12}$

(d)

${20 \choose 12} \cdot 2^{50} + {50 \choose 12} \cdot 2^{20}$

Solution

Let B be the set with exactly 12 beer bottles and any number of cider bottles
First, choose 12 beer bottles out of the 20: $ \binom{20}{12} $
We need to take into account all subsets of cider bottles that we add to the 12 beer bottles: $ 2^{50} $
$ |B| = \binom{20}{12} \cdot 2^{50} $
Let C be the set with exactly 12 cider bottles and any number of beer bottles
First, choose 12 cider bottles out of the 50: $ \binom{50}{12} $
We need to take into account all subsets of beer bottles that we add to the 12 cider bottles: $ 2^{20} $
$ |C| = \binom{50}{12} \cdot 2^{20} $
Let's determine $ B \cap C $: the set with exactly 12 beer bottles and 12 cider bottles
We need to choose 12 beer bottles and 12 cider bottles
First, choose 12 beer bottles out of the 20: $ \binom{20}{12} $
First, choose 12 cider bottles out of the 50: $ \binom{50}{12} $
$ |B \cap C| = \binom{20}{12} \cdot \binom{50}{12} $

Now, we can find $ B \cup C $

$ |B \cup C| = |B| + |C| - |B \cap C| $

$ |B \cup C| = \binom{20}{12} \cdot 2^{50} + \binom{50}{12} \cdot 2^{20} - \binom{20}{12} \cdot \binom{50}{12} $