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2024 AMC 10B Problem 22

Problem 22 of 25HarderNumber 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

Split 1616 people into 44 indistinguishable groups of 44 in 16!(4!)4 4!\dfrac{16!}{(4!)^4 \, 4!} ways, then each committee picks a chairperson and a secretary in 4⋅3=124 \cdot 3 = 12 ways, a factor of 124.12^4. Now count factors of 3.3. In 16!16! there are ⌊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+1=5;4 \cdot 1 + 1 = 5; and 12412^4 contributes 4.4. The exponent is 6−5+4=5,6 - 5 + 4 = 5, so r=5.r = 5. Therefore, the answer is A.
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Tagged: combinations · Legendre’s Formula · prime factorization

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