Difference between revisions of "2016 AMC 12A Problems/Problem 11"
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Our answer is <math>ab + bc + ac = 100 - (a + b + c) = 100 - 36 = \boxed{\textbf{(E)}\; 64}</math> | Our answer is <math>ab + bc + ac = 100 - (a + b + c) = 100 - 36 = \boxed{\textbf{(E)}\; 64}</math> | ||
+ | |||
+ | ==Solution 2== | ||
+ | An easier way to solve the problem: | ||
+ | Since <math>42</math> students cannot sing, there are <math>100-42=58</math> students who can. | ||
+ | |||
+ | Similarly <math>65</math> students cannot dance, there are <math>100-65=35</math> students who can. | ||
+ | |||
+ | And <math>29</math> students cannot act, there are <math>100-29=71</math> students who can. | ||
+ | |||
+ | Therefore, there are <math>58+35+71=164</math> students in all ignoring the overlaps between <math>2</math> of <math>3</math> talent categories. | ||
+ | There are no students who have all <math>3</math> talents, nor those who have none <math>(0)</math>, so only <math>1</math> or <math>2</math> talents are viable. | ||
+ | |||
+ | Thus, there are <math>164-100=\boxed{\textbf{(E) }64}</math> students who have <math>2</math> of <math>3</math> talents. | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2016|ab=A|num-b=10|num-a=12}} | {{AMC12 box|year=2016|ab=A|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:05, 19 October 2016
Contents
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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