2024 AMC 10B Problem 22

Problem 22 of 25HarderNumber 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

Split

16

people into

4

indistinguishable groups of

4

\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.

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,

r = 5.

Therefore, the answer is A.

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