Solution: 2022 Winter Final - 4

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 at least $10$ letters $a$?

(a)

$\sum_{i=10}^{70} \binom{70}{i}\cdot 5^{70-i}$

(b)

$5^{70} - \sum_{i=0}^{9} \binom{70}{i}\cdot 4^{70-i}$

(c)

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

(d)

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

(e)

None of the above

Solution

a is wrong because it only counts the number of strings with exactly 10 letters $a$

b is wrong because each repetition repeats saying, \enquote{we hava exactly i a’s} BUT it also says that the remaining positions can also be a since it lists all 5 characters being possible candidiates for each remaining position

d is wrong because it only counts the numbre of strings with exactly 10 letter $a$‘s. You forgot about whith exatly 11 a’s, 12 a’s, and so on

c is correct because it identifies all possibilities and subtracts the ones that don’t have at least 10 a’s