The US Senate consists of 100 members. Senate committees are to be formed so that each of the committees contains the same number of senators and each senator is a member of exactly one committee. The committees are to have more than 2 members but fewer than 50 members. There are various ways that these committees can be formed. a) What size committees are possible? b) How many committees are there for each size?
The US Senate consists of 100 members. Senate committees are to be formed so that each of the committees contains the same number of senators and each senator is a member of exactly one committee. The number of members of each committee is to be more than 2 but less than 50.
If each committee has N members, 100 divided by N has to be a whole number as no senator can be in more than 1 committee. The smallest number of members in a committee is 4. The number after 4 that 100 is divisible by is 5. This is followed by 10, 20 and 25.
There are 25 committees of 4 members each, 20 committees of 5 members each, 10 committees of 10 members each, 5 committees of 20 members each and 4 committees of 25 members each.
Essentially, the answer to both these questions rests in the factors of 100. We know that the number of committees and the number of senators in the committee have to be integers. Since there isn't any "double-counting" (senators only belong to one committee), this greatly eases the problem by just allowing us to divide the 100 senators into whatever number of committees to find the number of senators per committee. As such, these two values - the number of committees and the number of senators per committee - will be factors 100 and each pair will multiply to equal 100. All we have to do then is find the factors of 100, which are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The question specifies that the member count in a committee is greater than 2 and less than 50, which means the remaining values for the member counts in the committees are: 4, 5, 10, 20, and 25. All we have to do is pair these values up with an integer so that the product will be 100. These will then be the pairs of what size committees are possible (part a) and how many committees there are for each size.
4 member committees - 25 of them
5 member committees - 20 of them
10 member committees - 10 of them
20 member committees - 5 of them
25 member committees - 4 of them