Question

A group of $16$ people will be partitioned into $4$ 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:

Split $16$ people into $4$ indistinguishable groups of $4$ in $\dfrac{16!}{(4!)^4 \, 4!}$ ways, then each committee picks a chairperson and a secretary in $4 \cdot 3 = 12$ ways, a factor of $12^4.$ Now count factors of $3.$ In $16!$ there are $\lfloor 16/3 \rfloor + \lfloor 16/9 \rfloor = 6;$ the denominator $(4!)^4 \, 4!$ contributes $4 \cdot 1 + 1 = 5;$ and $12^4$ contributes $4.$ The exponent is $6 - 5 + 4 = 5,$ so $r = 5.$ Therefore, the answer is A.

2024 AMC 10B Problem 22

Solution:

Problem 22 in Other Years