Track / Stage 4 / 224 of 340 #484 of 1964

Problem 484

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Find the answer

I1.4 If $S$ is the number of ways to arrange $R$ persons in a circle, find the value of $S$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Reference: 1998 FI5.3, 2000 FG4.4
First arrange the 8 persons in a row. Number of permutations $=P_{8}^{8}=8!$
Suppose the first and the last in the row are $A$ and $H$ respectively.
Now join the first and the last persons to form a ring.
$A$ can be in any position of the ring. Each pattern is repeated 8 times.
$\therefore$ Number of permutations $=\frac{8!}{8}=5040$
Remark: the original version was ..."arrange $R$ people" ...
Note that the word "people" is an uncountable noun, whereas the word "persons" is a countable noun.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.