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 $r?$

$5$

$6$

$7$

$8$

$9$

Answer: A

Concepts:combinations Legendre’s Formula prime factorization

Difficulty rating: 2120

Small Hint:

The number of ways is $\dfrac{16!}{(4!)^4 \cdot 4!} \cdot (4 \cdot 3)^4,$ the partition count times chair/secretary choices per committee

Big Hint:

Count factors of $3$ in each piece using Legendre’s formula on $16!$ and $4!$

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 $\lfloor \frac{16}{3} \rfloor + \lfloor \frac{16}{9} \rfloor = 6;$ 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

Answer: A

Difficulty rating: 2120

Small Hint:

Big Hint:

Solution:

Problem 22 in Other Years