Problem:
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 ?
Answer Choices:
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Solution:
The 16 people can be partitioned into the 4 committees, each of size 4, in
ways; four of the 4 ! factors come from permuting the members of the committees and one 4 ! factor comes from permuting the committees. Then there are ways to choose the four chairpersons and ways to choose the four secretaries. This gives a total of
assignments. The numerator has 1 factor of 3 in each of , and 3 ; it has 2 factors of 3 in 9 and 4 factors in , a total of 10 . The denominator has 5 factors of 3 . Thus .
There are ways to choose the chairperson and secretary of the first committee, ways to choose the chairperson and secretary of the second committee, ways to choose the chairperson and secretary of the third committee, and ways to choose the chairperson and secretary of the fourth committee. There are then ways to choose the remaining members of the first committee, ways to choose the remaining members of the second committee, ways to choose the remaining members of the third committee, and way to choose the remaining members of the fourth committee. There are 4! ways to account for the fact that the committees are indistinguishable. This gives a total of
assignments. There are factors of 3 in the numerator and 1 factor of 3 in the denominator, so .
Note: The number of ways to make the assignments is .
The problems on this page are the property of the MAA's American Mathematics Competitions