Venn Diagram - 3

[ankitkumar13]

Q) A club with a total membership of 30 has formed 3 committees, M, S, and R, which have
8, 12, and 5 members, respectively. If no member of committee M is on either of the
other 2 committees, what is the greatest possible number of members in the club who are
on none of the committees?

Ans. - A) 5; B) 7; C) 8; D) 10; E) 12

Tried applying the formula for Venn Diagram (3 sets), but just not able to get the right answer........ Pls help

[gokul_nair1984]

Draw 3 circles and try to analyze it logically.

M=8(only)

Therefore, 22 remain(30-8)

R and S can have members in common or they might not have members in common.

To maximize members who are on none of the committees, we will have to minimize members in R&S.

The maximum no of people are not in any committee when all the 5 members of R are the members of S as well. Therefore consider all the members of R to be a part of S.

=>n (S ∪ R) = n (S) = 12.

Therefore, The maximum no of people are not in any committee is 22-12=10

[RonPurewal]

gokul, this is the correct solution, although i'll clarify some of the wording:

Draw 3 circles and try to analyze it logically.

M=8(only)

Therefore, 22 remain(30-8)

if you draw a venn diagram, the condition stated in the problem means that three of the regions will contain a zero:
* the intersection between committees M and R should contain a zero.
* the intersection between committees M and S should contain a zero.
* the intersection of all three committees should contain a zero.

therefore, once you've drawn in all those zeros (and the 8 that goes in the region corresponding to committee M alone), you should find that it reduces to a problem of just two committees -- R and S -- into which you have to distribute the remaining 22 people.

R and S can have members in common or they might not have members in common.

To maximize members who are on none of the committees, we will have to minimize members in R&S.

this is correct. however, to clarify:
we want to minimize the total number of individuals represented in the three regions corresponding to committees R and S --
* the region corresponding to committee R by itself;
* the region corresponding to committee S by itself; and
* the region at the intersection of committees R and S.

in order to minimize this total number, you have to MAXIMIZE the number of individuals in the intersection (since, by putting someone in the intersection, you only have to include one person in order to be counted in both committees -- as opposed to the two people you would have to put into the committees separately).

The maximum no of people are not in any committee when all the 5 members of R are the members of S as well. Therefore consider all the members of R to be a part of S.

=>n (S ∪ R) = n (S) = 12.

Therefore, The maximum no of people are not in any committee is 22-12=10

yep.