2024 AMC 12B Problems/Problem 16

The following problem is from both the 2024 AMC 10B #22 and 2024 AMC 12B #16, so both problems redirect to this page.

Problem 16

A group of people will be partitioned into indistinguishable -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 . What is ?

Solution

Solution 1

There are ways to choose the first committee, ways to choose the second, for the third, and for the fourth. Since the committees are indistinguishable, we need to divide the product by . Thus the people can be grouped in ways.

In each committee, there are ways to choose the chairperson and secretary, so ways for all committees. Therefore, there are total possibilities.

Since contains factors of , contains , and contains , .

~kafuu_chino

Solution 2 (Multinomial Coefficients)

There are ways to choose the 4 committees. You have to divide by another 4! since the order of the committees does not matter. Furthermore, in each committee, you can have ways to choose chairperson and secretary. Hence a total of

~mathboy282

Solution 3

There will be ways to pick the chairperson of the first committee, then to pick the secretary, and lastly ways to pick the other two members of the first committee. Similarly, we can complete the rest of the terms as follows: We notice the numerator has at most , and the denominator has just . Thus, the value of in question is .

~lisztepos

Solution 4

We can convert the question into arranging people in a line of length .

First, arrange all people in a line. There are possible ways to do this. Then, divide the line into groups of , which represent the committees. Since the committees are indistinguishable, we divide by the number of ways to permute the groups, .

Within each group of , the first two positions in the line are assigned as the chairperson and the secretary, while the remaining two are ordinary members. Since the roles of chairperson and secretary are distinct, but the order of the remaining two members does not matter, we divide by for each group. Thus, we divide by to account for all committees.

The total number of ways to assign people into committees with roles is given by: (This is equivalent to Solution 1's .)

To find , we use Legendre's Formula to count the total power of in : Each contributes one factor of , and since we divide by a total of times, we subtract from : Thus, the value of is . ~ryk

Video Solution 1 by Pi Academy (In A Lot More Than 2 Mins ⚡🚀)

https://youtu.be/9ymwnHnTbDQ?feature=shared

~ Pi Academy

Video Solution 2 by Innovative Minds

https://youtu.be/HMPHdBiaYQc

Video Solution 3 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

Video Solution 4 by sevenblade(standard approach!)

https://www.youtube.com/watch?v=5BXclh_DLEg

Fast Solution

https://www.youtube.com/watch?v=jPTL8hf0Ur0&t=1s

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.