Question: 2018 Winter Final - 3

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 at least 3 beer bottles (and any number of cider bottles). How many such subsets are there?

(a)

$2^{70} - 2^{50} - 20 \cdot 2^{50}$

(b)

$2^{70} - 2^{50} - 20 - {20 \choose 2}$

(c)

$2^{70} - 2^{50} - 20 \cdot 2^{50} - {20 \choose 2} \cdot 2^{50}$

(d)

None of the above.