At a certain high school, the Prom Committee is going to choose new members. There are 4 students from the Junior class and 8 students from the Senior class who are willing to be new members. In how many ways can 5 new members be chosen if more than 3 must be from the Senior class?(If necessary, consult a list of formulas.)
Question
At a certain high school, the Prom Committee is going to choose new members. There are 4 students from the Junior class and 8 students from the Senior class who are willing to be new members. In how many ways can 5 new members be chosen if more than 3 must be from the Senior class?(If necessary, consult a list of formulas.)
Solution
To solve this problem, we need to use the combination formula which is C(n, r) = n! / [r!(n-r)!], where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.
The problem states that more than 3 members must be from the Senior class. This means we can have 4 or 5 members from the Senior class.
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If 4 members are chosen from the Senior class, 1 member is chosen from the Junior class. The number of ways to do this is C(8, 4) * C(4, 1).
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If all 5 members are chosen from the Senior class, no members are chosen from the Junior class. The number of ways to do this is C(8, 5) * C(4, 0).
Adding these two cases together gives the total number of ways to choose the committee.
Let's calculate:
- C(8, 4) * C(4, 1) = (8! / 4!(8-4)!) * (4! / 1!(4-1)!) = 70 * 4 = 280
- C(8, 5) * C(4, 0) = (8! / 5!(8-5)!) * (4! / 0!(4-0)!) = 56 * 1 = 56
So, the total number of ways to choose the committee is 280 + 56 = 336.
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