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Problem 666

AIME late

Combinatorics Difficulty 5.2 Find the answer

11. String 6 red balls, 1 white ball, and 8 yellow balls into a necklace, then the number of possible arrangements is $\qquad$ (balls of the same color are indistinguishable).

Your answer

Official solution

11. 1519 Fix the white ball, then the necklace arrangement becomes a linear arrangement (but note that it can still be flipped), with a total of $\frac{(6+8)!}{6!8!}=3003$ ways, of which the left-right symmetric linear arrangements are $\frac{7!}{3!4!}=35$ ways. Therefore, there are $\frac{2968}{2}+35=1519$ ways.

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