AMC 10B/12B 2024 (Problem 22/16)
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 , where and are positive integers and is not divisible by 3. What is ?
The number of ways to split 16 people into 4 indistinguishable 4-person committees is
For each committee, we must select a chairperson and a secretary. There are ways to do this for each committee. Thus,
Expanding the binomial coefficients,
Simplifying,
Since and none of the other factors is divisible by , the highest power of dividing this product is
Therefore,
Expanding the binomial coefficients,
Simplifying,
Since and none of the other factors is divisible by , the highest power of dividing this product is
Therefore,
(A) 5