Question: 2022 Winter Final - 3
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 $a$ or exactly $15$ letters $b$?
(a)
$\binom{70}{5}\cdot 4^{65} + \binom{70}{15}\cdot 4^{55}$
(b)
$\binom{70}{5}\cdot 4^{65} + \binom{70}{15}\cdot 4^{55} - \binom{70}{5}\cdot\binom{65}{15}\cdot 3^{50}$
(c)
$\binom{70}{5}\cdot 4^{65} + \binom{65}{15}\cdot 4^{50} $
(d)
None of the above
(e)
$\binom{70}{5}\cdot 4^{65} + \binom{65}{15}\cdot 4^{50} - \binom{70}{5}\cdot\binom{65}{15}\cdot 3^{50}$