UNO High School Problem of the Week Competition
Problem 1
It's the beginning of the school year and the student government needs to select a president, vice president, secretary, and treasurer. How many possible ways are there to fill these positions if there are 12 candidates?
Suppose on the other hand, that the student government wants a committee of size 4 (order doesn't matter). How many different committees can be formed if there are still 12 candidates?
Solution
Since the order matters in the first problem, it is a permutation. The formula for solving the number of permutations of a subset of r elements from a set containing n elements is the following:
P(n, r) = n!/(n-r)!
Substituting 12 for n and 4 for r, we get:
P(12, 4) = 12!/(12-4)! = 11, 880 different ways to choose a president, vice president, secretary, and treasurer.
Since the order does not matter in the second problem, it is a combination. The formula for solving the number of combinations of a subset of r elements from a set containing n elements is the following:
C(n, r) = n!/((n-r)!r!)
Substituting 12 for n and 4 for r, we get:
C(12,4) = 12!/((12-4)!4!) = 495 different ways to form a four person committee