2021 AMC 10B Problem 22

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 $$\begin{aligned} &\binom51(4!)^2-\binom52(3!)^2 \\ &\quad {}+\binom53(2!)^2-\binom54(1!)^2 \\ &\quad {}+\binom55(0!)^2. \end{aligned}$$ 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 .