Solution: 2018 Fall Midterm - 8

Author: Michiel Smid

Question

Consider a set $\mathcal{B} = \{B_1,B_2,\dots,B_{13}\}$ of 13 beer bottles and a set $\mathcal{C} = \{C_1,C_2,\dots,C_{12}\}$ of 12 cider bottles.
Consider subsets $X$ of $\mathcal{B} \cup \mathcal{C}$, such that $X$ consists of exactly 5 beer bottles and all cider bottles in $X$ have an even index.
How many such subsets $X$ are there?

(a)

None of the above.

(b)

${13 \choose 5} \cdot 2^6$

(c)

${13 \choose 5} \cdot 2^5$

(d)

${12 \choose 5} \cdot 2^6$

Solution

We need to choose 5 beer bottles from the set $ {B_1, B_2, …, B_{13}} $

There are $ \binom{13}{5} $ ways to choose 5 beer bottles from the set $ {B_1, B_2, …, B_{13}} $

We can have 0 to 6 cider bottles in the subset $ X $

We take into account all possible numbers of cider bottles in $ X $ using the size of sets method: $2^6$

Thus, there are $ \binom{13}{5} \cdot 2^6 $ such subsets.