Question

Each of $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\dfrac{p}{q}?$

$1$

$4$

$8$

$12$

$16$

Answer: E

Concepts:combinations basic probability

Difficulty rating: 1540

Solution:

All $5^{20}$ assignments of the distinguishable balls to the labeled bins are equally likely. For $q,$ the number of assignments is $\frac{20!}{(4!)^5}.$

For $p,$ choose the bin with $3$ balls and the bin with $5$ balls in $5\cdot4$ ways. The number of assignments is then $5\cdot4\cdot\frac{20!}{3!5!(4!)^3}.$

The common probability denominator cancels, so $\frac pq=20\cdot\frac{(4!)^2}{3!5!}=20\cdot\frac45=16.$

Thus, E is the correct answer.

2021 AMC 10A Fall Problem 21

Answer: E

Difficulty rating: 1540

Solution:

Problem 21 in Other Years