Solution: 2018 Fall Final - 2

Author: Michiel Smid

Question

Consider strings of length 70, in which each character is one of the letters $a, b, c$. How many such strings have exactly 12 letters $c$ and exactly 30 letters $b$?

(a)

${70 \choose 12} \cdot {58 \choose 30}$

(b)

${70 \choose 12} \cdot {58 \choose 30} \cdot 3^{28}$

(c)

${70 \choose 12} + {58 \choose 30}$

(d)

${70 \choose 12} \cdot {58 \choose 30} \cdot 2^{28}$

Solution

Well, we first choose 12 positions out of the 70 for the letter $c$: $ \binom{70}{12} $

Then, we choose 30 positions out of the remaining 58 for the letter $b$: $ \binom{58}{30} $

The remaining 28 positions are for the letter $a$: 1

$ |C| = \binom{70}{12} \cdot \binom{58}{30} $