Solution: 2018 Fall Midterm - 4

Author: Michiel Smid

Question

Elisa Kazan's neighborhood pub serves 8 different types of cider; denote these types by $C_1,C_2,\dots,C_8$. Elisa invites 7 friends to this pub and orders one cider for each friend. Different friends may get the same type of cider. (Elisa came by car and, therefore, orders a glass of water for herself.)
In how many ways can Elisa place these orders of cider, such that exactly 4 of her friends get a cider of type $C_3$?

(a)

${7 \choose 4} \cdot 7^4$

(b)

${7 \choose 4} \cdot 8^3$

(c)

${7 \choose 4} \cdot 8^4$

(d)

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

Solution

We choose 4 friends to get cider of type $ C_3 $ from the 7 friends: $ \binom{7}{4} $

The remaining friends can get cider of any of the other 7 types: $ 7^3 $

Thus, there are $ \binom{7}{4} \cdot 7^3 $ ways to place these orders.