Solution: 2018 Winter Midterm - 4

Author: Michiel Smid

Question

Let $b \geq 1$ and $c \geq 1$ be integers. Elisa Kazan's neighborhood pub serves $b$ different types of beer and $c$ different types of cider. Elisa invites 6 friends to this pub and orders 7 drinks, one drink (beer or cider) for each friend, and one cider for herself. Different people may get the same type of beer or cider.
In how many ways can Elisa place these orders, such that exactly 4 people get a beer?

(a)

${6 \choose 4} \cdot b^{4} \cdot c^{2}$

(b)

${6 \choose 4} \cdot b^{4} \cdot c^{3}$

(c)

${7 \choose 4} \cdot b^{4} \cdot c^{3}$

(d)

None of the above.

Solution

We choose 4 friends to get beer from the 6 friends: $ \binom{6}{4} $

We give each of the 4 friends a beer of b types: $ b^4 $

We give the 3 remaining friends a cider of c types: $ c^3 $

Thus, there are $ \binom{6}{4} \cdot b^4 \cdot c^3 $ ways to place these orders.