2013 AMC 10A Problems/Problem 11
Problem
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?
Solution
Let the number of students on the council be . To select a two-person committee, we can select a "first person" and a "second person." There are
choices to select a first person; subsequently, there are
choices for the second person. This gives a preliminary count of
ways to choose a two-person committee. However, this accounts for the order of committees. To understand this, suppose that Alice and Bob are two students in the council. If we choose Alice and then Bob, that is the same as choosing Bob and then Alice; however in our counting, the latter and former arrangements would be considered the same. Therefore, we have to divide by
to account for overcounting. Thus, there are
ways to choose the two-person committee. Solving this equation, we find that
and
are integer solutions.
is a ridiculous situation, so there are
people on the student council. The solution is
.
Solution 2 (much faster)
To choose people from
total people and get
as a result, we can establish the equation
which we can easily see is 5. The question asks how many ways to choose
people, so there are
ways.
See Also
2013 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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