2024 AMC 12B Problem 16

Problem 16 of 25IntermediateNumber Theory Counting & Probability

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?

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Solution

The number of ways to split

16

people into

4

indistinguishable groups of

4

\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 \frac{16}{3}\rfloor + \lfloor \frac{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.

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