Question

Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$

$47$

$94$

$227$

$471$

$542$

Video solution:

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Written solution:

Fix Ang's placement and label each box by the color Ang put in it. Ben and Jasmin each choose a permutation of the five colors, so there are $(5!)^2$ equally likely pairs of placements.

For a specified set of $k$ boxes to receive three blocks of the same color, both Ben and Jasmin must match Ang in those $k$ boxes. This can happen in $((5-k)!)^2$ ways. By inclusion-exclusion, the number of successful placement pairs is

$\binom51(4!)^2-\binom52(3!)^2+\binom53(2!)^2-\binom54(1!)^2+inom55(0!)^2.$

This equals

$2880-360+40-5+1=2556.$

Therefore the probability is

$\frac{2556}{(5!)^2}=\frac{2556}{14400}=\frac{71}{400}.$

Thus $m+n=71+400=471$ .

Thus, the answer is D .

2021 AMC 10B Spring Problem 22

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Written solution:

Problem 22 in Other Years