Solution: 2022 Winter Final - 2

Author: Michiel Smid

Question

Consider strings of length $70$, in which each character is one of the characters $a,b,c,d,e$. How many such strings have exactly $5$ letters $a$ and exactly $15$ letters $b$?

(a)

$\binom{70}{5}\cdot\binom{70}{15}\cdot 3^{50}$

(b)

$\binom{70}{5}\cdot\binom{65}{15}\cdot 3^{50}$

(c)

$\binom{70}{15}\cdot\binom{55}{5}\cdot 5^{50}$

(d)

$\binom{70}{5}\cdot\binom{65}{15}\cdot 5^{50}$

(e)

$\binom{70}{5}\cdot\binom{65}{15}\cdot 4^{50}$

Solution

First, we choose 5 positions out of the 70 positions to place $a$‘s: $ \binom{70}{5} $

Then, we choose 15 positions out of the remaining 65 positions to place $b$‘s: $ \binom{65}{15} $

Each of the remaining 50 positions can be any of the 3 other letters: $ 3^{50} $

Thus, the final answer is $ \binom{70}{5} \cdot \binom{65}{15} \cdot 3^{50} $