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MartyMurray
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In a show, there are 34 performers, all of whom dance, sing, or play Updated on: 07 Mar 2026, 06:15
Originally posted by MartyMurray on 23 Mar 2025, 13:38.
Last edited by MartyMurray on 07 Mar 2026, 06:15, edited 1 time in total.
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In a show, there are 34 performers, all of whom dance, sing, or play a musical instrument. If 20 of the performers dance, 26 sing, and 10 play a musical instrument, what is the minimum number of them who do exactly two of those things?

(A) 0

(B) 2

(C) 10

(D) 11

(E) 22


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MartyMurray
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In a show, there are 34 performers, all of whom dance, sing, or play 23 Mar 2025, 17:45
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In a show, there are 34 performers, all of whom dance, sing, or play a musical instrument. If 20 of the performers dance, 26 sing, and 10 play a musical instrument, what is the minimum number of them who do exactly two of those things?

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(those who do all 3)

To minimize the number of them who do exactly 2, maximize the number who do all 3.

Since 10 play a musical instrument, the maximum number who can do all 3 is 10.

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(10)

34 = 36 - (those who do exactly 2)

(those who do exactly 2) = 2

(A) 0

(B) 2

(C) 10

(D) 11

(E) 22

Correct answer: B
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In a show, there are 34 performers, all of whom dance, sing, or play 01 Jun 2025, 05:43
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Thanks for this simple explanation. I tried building out the Venn diagram and solved it by the longer method. This method is much more simpler :)
MartyMurray
In a show, there are 34 performers, all of whom dance, sing, or play a musical instrument. If 20 of the performers dance, 26 sing, and 10 play a musical instrument, what is the minimum number of them who do exactly two of those things?

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(those who do all 3)

To minimize the number of them who do exactly 2, maximize the number who do all 3.

Since 10 play a musical instrument, the maximum number who can do all 3 is 10.

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(10)

34 = 36 - (those who do exactly 2)

(those who do exactly 2) = 2

(A) 0

(B) 2

(C) 10

(D) 11

(E) 22

Correct answer: B
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In a show, there are 34 performers, all of whom dance, sing, or play 02 Jun 2025, 01:52
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Hi MartyMurray
Thank you for the solution, I tried the same method and in the end i maximized the people who do all 3 things to 11, and that gave me the answer as 0. My question is why can't we maximize the people who do all 3 things to 11 instead of maximizing to 10 as you did ?
MartyMurray
In a show, there are 34 performers, all of whom dance, sing, or play a musical instrument. If 20 of the performers dance, 26 sing, and 10 play a musical instrument, what is the minimum number of them who do exactly two of those things?

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(those who do all 3)

To minimize the number of them who do exactly 2, maximize the number who do all 3.

Since 10 play a musical instrument, the maximum number who can do all 3 is 10.

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(10)

34 = 36 - (those who do exactly 2)

(those who do exactly 2) = 2

(A) 0

(B) 2

(C) 10

(D) 11

(E) 22

Correct answer: B
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In a show, there are 34 performers, all of whom dance, sing, or play 02 Jun 2025, 03:53
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IMpratik
Hi MartyMurray
Thank you for the solution, I tried the same method and in the end i maximized the people who do all 3 things to 11, and that gave me the answer as 0. My question is why can't we maximize the people who do all 3 things to 11 instead of maximizing to 10 as you did ?
MartyMurray
In a show, there are 34 performers, all of whom dance, sing, or play a musical instrument. If 20 of the performers dance, 26 sing, and 10 play a musical instrument, what is the minimum number of them who do exactly two of those things?

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(those who do all 3)

To minimize the number of them who do exactly 2, maximize the number who do all 3.

Since 10 play a musical instrument, the maximum number who can do all 3 is 10.

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(10)

34 = 36 - (those who do exactly 2)

(those who do exactly 2) = 2

(A) 0

(B) 2

(C) 10

(D) 11

(E) 22

Correct answer: B
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We can't maximize people who can do all 3 to 11 is because there are only 10 people who can play a musical instrument. When you say 11 people can do all 3, it also means that 11 people can play musical instrument which is not true. So maximum number of people who do all 3 cannot be greater than 10.
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In a show, there are 34 performers, all of whom dance, sing, or play 02 Jun 2025, 06:55
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Another way to think about it is that the number of people who do all three things, cannot be larger than any of 10, 20, or 26. Since 10 is the smallest of those three, then it cannot be bigger than 10
IMpratik
Hi MartyMurray
Thank you for the solution, I tried the same method and in the end i maximized the people who do all 3 things to 11, and that gave me the answer as 0. My question is why can't we maximize the people who do all 3 things to 11 instead of maximizing to 10 as you did ?
MartyMurray
In a show, there are 34 performers, all of whom dance, sing, or play a musical instrument. If 20 of the performers dance, 26 sing, and 10 play a musical instrument, what is the minimum number of them who do exactly two of those things?

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(those who do all 3)

To minimize the number of them who do exactly 2, maximize the number who do all 3.

Since 10 play a musical instrument, the maximum number who can do all 3 is 10.

34 = 20 + 26 + 10 - (those who do exactly 2) - 2(10)

34 = 36 - (those who do exactly 2)

(those who do exactly 2) = 2

(A) 0

(B) 2

(C) 10

(D) 11

(E) 22

Correct answer: B
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