Question: 2018 Winter Final - 2

Author: Michiel Smid

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)

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

(b)

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

(c)

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

(d)

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