Question

A group of $16$ people will be partitioned into indistinguishable $4$ -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$

$5$

$6$

$7$

$8$

$9$

Solution:

The number of ways to split $16$ people into $4$ indistinguishable groups of $4$ is $\dfrac{16!}{(4!)^4\, 4!}.$ Each committee then chooses a chairperson and a secretary in $4 \cdot 3 = 12$ ways, contributing $12^4.$ So the total is $\dfrac{16!}{(4!)^4\,4!}\cdot 12^4.$

Counting factors of $3:$ $16!$ contributes $\lfloor 16/3\rfloor + \lfloor 16/9\rfloor = 6.$ The denominator $(4!)^4\,4!$ contributes $4 + 1 = 5.$ And $12^4 = (2^2\cdot 3)^4$ contributes $4.$ Thus $r = 6 - 5 + 4 = 5.$

Thus, the correct answer is A.

2024 AMC 12B Problem 16

Solution:

Problem 16 in Other Years