I don't know how to describe this problem

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4 friends were singing at the concert. Each song was performed by 3 of them. Person A performed the most --> 10 songs, person B performed the least --> 8 songs. In total, how many songs did friends perform?
Answers: 1) 8; 2) 12; 3) 10; 4) 11.

I've been thinking for some time but I've got no idea how to tackle this problem. Describe your thought process while solving, please.
 
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Aug 2007
3,171
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USA
4 friends were singing at the concert. Each song was performed by 3 of them. Person A performed the most --> 10 songs, person B performed the least --> 8 songs. In total, how many songs did friends perform?
Answers: 1) 8; 2) 12; 3) 10; 4) 11.

I've been thinking for some time but I've got no idea how to tackle this problem. Describe your thought process while solving, please.
#1 - Throw out 8. "A" sang 10. It can't be 8.
#2 - What did "C" and "D" sing? "Least" and "Most" imply no ties. Otherwise, superlatives don't work. "C" and "D" must have sung how many songs?

Are we getting anywhere?
 
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Aug 2006
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4 friends were singing at the concert. Each song was performed by 3 of them. Person A performed the most --> 10 songs, person B performed the least --> 8 songs. In total, how many songs did friends perform?
Answers: 1) 8; 2) 12; 3) 10; 4) 11.

I've been thinking for some time but I've got no idea how to tackle this problem. Describe your thought process while solving, please.
Here is my solution:
$\dbinom{4}{3}=4$ Four choose three is four. So there are four groups of three.

$ABC\quad 3\\ABD\quad 3\\ACD\quad 4\\BCD\quad 2$

Now I leave to you to see that the above meets all requirements.

I did that a Venn diagram on four sets.
 
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Jun 2013
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songs.png

\(\displaystyle 3n=10+8+9+9\)