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2024 AMC 12B Problem 16

Problem 16 of 25IntermediateNumber TheoryCombinatorics

A group of 1616 people will be partitioned into 44 indistinguishable 44-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as 3rM,3^r M, where rr and MM are positive integers and MM is not divisible by 3.3. What is r?r?

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Solution

The number of ways to split 1616 people into 44 indistinguishable groups of 44 is 16!(4!)4 4!.\dfrac{16!}{(4!)^4\, 4!}. Each committee then chooses a chairperson and a secretary in 4⋅3=124 \cdot 3 = 12 ways, contributing 124.12^4. So the total is 16!(4!)4 4!⋅124.\dfrac{16!}{(4!)^4\,4!}\cdot 12^4. Counting factors of 3:3: 16!16! contributes ⌊163⌋+⌊169⌋=6.\lfloor \frac{16}{3}\rfloor + \lfloor \frac{16}{9}\rfloor = 6. The denominator (4!)4 4!(4!)^4\,4! contributes 4+1=5.4 + 1 = 5. And 124=(22⋅3)412^4 = (2^2\cdot 3)^4 contributes 4.4. Thus r=6−5+4=5.r = 6 - 5 + 4 = 5. Thus, the correct answer is A.
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