Difficult Math Problem #93 - Sets

Assuming all the 5 pl in club Z also joined club S. It would 30-10+7+5 = 8
Am I right?

oa:

total no of people = 30
no joining M = 10
no joining S = 12
no joining Z = 5
question asked - AT MOST how many people did not join any group?


solution: now since none of the members of M joined any other group, the no of people left = 30-10(for M)=20
since the question says at most how many did not join any group, lets assume the all people who join Z also join S. so no of people joining group S and Z are 12 (note that there will be 5 people in group S who have also joined Z)

therefore no of people not joining any group = 20-12=8
Hence e

I think if you make a diagram it will be explained much better.If you make a diagram and since M=10 does not share with any all the intersections containing 10 will be zero and assume intersection of S and Z be X and so Z only will be Z-X or 5-X and other will be S-X or 12-X
Since Z only cant be negative so max value X can take is 5 So X will be 5 and Sum will be 10+7+5=22 therefore no of people not joining any group = 30-22=8

I made a Venn diagram to solve this:

Thanks buddy

The most important information is that the members of club M did not join any other club. This gives you the information that 0 members join M & S, M & Z and M & S & Z.