# How many of the groups do chess and drama but not art and how many of the groups belong to the chess club from the following information:All the members of a group of 30 teenagers belong to at...

How many of the groups do chess and drama but not art and how many of the groups belong to the chess club from the following information:

All the members of a group of 30 teenagers belong to at least one club. There are 3 clubs, chess, drama and art.the data is as follows. 6 of the teenagers belong to only the art club. 5 of the teenagers belong to all 3 clubs. 2 of the teenagers belong to the chess and art clubs but not to the drama club. 15 of the teenagers belong to the art club. 2 of the teenagers belong only to the chess club. 3 of the teenagers belong only to the drama club.

Asked on by piqachu

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

There are 30 teenagers and each of them is in at least one of the three clubs. The chess club is represented by C, the drama club by D and the art club by A.

15 of the teenagers belong to the art club

=> `n(A) = 15`

5 of the teenagers belong to all 3 clubs.

=> `n(AnnCnnD) = 5`

Six of the teenagers belong to only the art club.

=> `n(A) - n(AnnC) - n(AnnD) + n(AnnCnnD)= 6`

=> `15 - n(AnnC) - n(AnnD) + 5 = 6`

=> `n(AnnC) + n(AnnD) = 14`

=> `7 + n(AnnD) = 14`

=> `n(AnnD) = 7`

2 of the teenagers belong only to the chess club

=> `n(C) - n(AnnC) - n(CnnD) + n(AnnCnnD) = 2`

=> `n(C) - 7 - n(CnnD) + 5 = 2`

=> `n(C) - n(CnnD) = 4`

3 of the teenagers belong only to the drama club

=> `n(D) - n(AnnD) - n(CnnD) + n(AnnDnnC) = 3`

=> `n(D) - n(CnnD) = 5`

2 of the teenagers belong to the chess and art clubs but not to the drama club

=> `n(AnnC) - n(AnnDnnC) = 2`

=> `n(AnnC) - 5 = 2`

=> `n(AnnC) = 7`

Also, `n(A) + n(C) + n(D) - n(AnnC) - n(AnnD) - n(CnnD) `

`+ n(AnnCnnD) = 30`

`15 + 4 + n(D) - 7 - 7 + 5 = 30`

=> `n(D) = 20`

`n(CnnD) = 15`

The number of teenagers that belong to the chess and drama clubs but not the art club is `n(CnnD) - n(AnnCnnD) = 15 - 5 = 10`

The number of teenagers in the chess club is `n(C) = 19`

The number of teenagers is the chess club is 19 and the number of teenagers that belong to the chess and drama clubs but not the art club is 10

piqachu | Student, Undergraduate | (Level 2) eNoter

Posted on

sorry for the incomplete question.

from the data above, the questions are:

(a)    How many of the groups do chess and drama but not art?

(b)    How many of the groups belong to the chess club?

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