Solution: 2022 Winter Final - 1

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 $10$ letters $e$?

(a)

$\binom{70}{5}\cdot 4^{60}$

(b)

$\binom{70}{5}\cdot 5^{60}$

(c)

$\binom{70}{10}\cdot 5^{60}$

(d)

$\binom{70}{10}\cdot 4^{60}$

(e)

$5^{10}\cdot 4^{60}$

First, we choose 10 positions out of the 70 positions to place $e$‘s: $ \binom{70}{10} $

Each of the remaining 60 positions can be any of the 4 other letters: $ 4^{60} $

Thus, the final answer is $ \binom{70}{10} \cdot 4^{60} $