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Problem 634

AIME late

Combinatorics Difficulty 5.1 Find the answer

8. To color the eight vertices of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with four different colors, such that the two endpoints of the same edge have different colors, there are a total of coloring methods.

Your answer

Official solution

8. 2652 .

First, color the four points $A, B, C, D$ above, with 84 coloring methods. Then consider the four points below, using the principle of inclusion-exclusion, the total number of methods is
$\begin{array}{l} 84\left\{84-C_{4}^{1}[3 \times(3+2 \times 2)+\right. \\ (3+2 \times 2)]+2\left(\frac{36}{84} \times 9+\frac{48}{84} \times 4\right)- \\ \left.C_{4}^{3}\left(\frac{36}{84} \times 3+\frac{48}{84} \times 2\right)+C_{4}^{4}\right\} \\ =2652 . \end{array}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.