2016 AMC 12A Problems/Problem 11
Contents
[hide]Problem
Each of the students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are students who cannot sing, students who cannot dance, and students who cannot act. How many students have two of these talents?
Solution
Let be the number of students that can only sing, can only dance, and can only act.
Let be the number of students that can sing and dance, can sing and act, and can dance and act.
From the information given in the problem, and .
Adding these equations together, we get .
Since there are a total of students, .
Subtracting these equations, we get .
Our answer is
Solution 2
An easier way to solve the problem: Since students cannot sing, there are students who can.
Similarly students cannot dance, there are students who can.
And students cannot act, there are students who can.
Therefore, there are students in all ignoring the overlaps between of talent categories. There are no students who have all talents, nor those who have none , so only or talents are viable.
Thus, there are students who have of talents.
See Also
2016 AMC 12A (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 12 Problems and Solutions |
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