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I am stuck with this question,

In a group of 42 theatrical performers, we have 21 singers, 22 actors and 21 dancers. There are some managers who cannot perform on stage. 10 people can sing and act. 10 people can dance and act. 8 people can sing and dance. 5 people can write poetry. All poets are actors, but 2 of them can dance too. 3 people can direct plays. All directors are dancers, but 1 can sing too. 3 people can sing, act and dance. Find the number of managers. How many people can do any two things together?

I made the Venn diagram, enter image description here

The problem I am facing is that that if I am finding just the actors, they come out to be negative so I am not sure about my result. I would be thankful if anyone could help me out in it. Thanks

EDIT 2:

Thank-you for the help. I am again drawing the Venn diagram. This time I am having a small problem in determining if A should contain 5(as Brain mentioned) or it should be 2 (this is also same as Brian mentioned but out of 5, 3 are in $P$ so it should be 2) right? enter image description here

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Your venn diagram is not correct. Try to draw it again. –  Hassan Muhammad Nov 3 '11 at 9:06

1 Answer 1

up vote 3 down vote accepted

Work out from the middle: you know that there are $3$ people who can sing, dance, and act. There are $10$ who can sing and act, but $3$ of them are already accounted for, so the shaded region at the top where you have $10$ should really have $7$: $7$ of the people who can sing and act cannot dance, and the other $3$ can. Similarly, there are $7$ who can act and dance but cannot sing, and there are $5$ who can dance and sing but cannot act.

Now consider the singers. There are $21$ of them altogether. $7$ of these can also act but cannot dance; $3$ can also act and dance; and $5$ can also dance but cannot act. These singers who can do something else as well account for $15$ of the $21$ singers, so there must be $6$ singers who can neither act nor dance. Similar reasoning, which I’ll leave you to try, shows that there are $6$ dancers who can neither sing nor act and $5$ actors who can neither dance nor sing.

If you now add up the figures in all seven regions within the three circles, you should get a total of $39$. Since there are $42$ people altogether, this means that there must be $42-39=3$ managers. Note that up to this point the directors and poets are red herrings: you can ignore them completely.

The last question is ambiguous. I can’t tell whether it wants the total number of people who can do exactly two things, the total number who can do at least two things, or for every pair of things the number who can do at least (or exactly) those two things. If it’s asking for the number of people who can do at least two things, we can start with the $3+7+7+5=22$ people in the various intersections of the circles. Now consider the poets: all of them are actors, so all of them can do at least two things. Two of them, however, are actor-dancers, so we’ve already counted them in the $22$. The other $3$, however, are actors who neither dance nor sing, so they weren’t counted and need to be added; this brings the total to $25$. Similarly, one of the $3$ directors has already been counted (as a dancer-singer), but the other two are dancers who neither sing nor act, so they have to be added to the total of those with multiple talents, bringing it to $27$.

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Thankyou so very much for the answer. Please check out the edit. –  Fahad Uddin Nov 3 '11 at 9:40
    
@Akito: The new diagram looks good. If you want to finish it off completely, you should have $4$ in the just dancer category and $3$ in the region outside the circles. –  Brian M. Scott Nov 3 '11 at 10:10
    
Thanks again. It went perfect. One thing I wanted to ask was that can we name the sets like $DANCERS$, $PLAYERS$ etc or is it necessary to have a single character name(like $P$, $Q$ etc)? –  Fahad Uddin Nov 3 '11 at 11:28
    
@Akito: When you start using the names of the sets in formulas, writing expressions like $S\cap D$, for instance, it’s conventional and much handier to have single-character names. If you’re just labelling regions in a Venn diagram, it doesn’t really matter. –  Brian M. Scott Nov 3 '11 at 22:45

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