Each vertex of convex polygon $ABCDE$ is to be assigned a color. There are $7$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
$\textbf{(A)}\ 3430 \qquad \textbf{(B)}\ 4200 \qquad \textbf{(C)}\ 6349 \qquad \textbf{(D)}\ 7770 \qquad \textbf{(E)}\ 8400$
Answer: D: 7770
We can use the PIE to solve this problem, the answer is $7^5-7^4\cdot 5+10\cdot 7^3-10 \cdot 7^2+ 5\cdot 7-7=7770$
All possible colorings are $7^5$ ways, remove vertices on one diagonal are the same, $5\cdot 7^4$, add back vertices on two diagonals are the same $10\cdot 7^3$,
remove vertices on three diagonals are the same $10\cdot 7^2$, add back vertices on 4 diagonals are the same $5\cdot 7$, remove all are the same $7$.
AMC 10/12 exam time:
A: Nov. 10, 2022
B: Nov 16, 2022
AMC 8: January, 17-23, 2023
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