I'm working on a following question:
Let $S$ be a set of students who solve maths problems. $P$ is the subset of $S$ containing all students who eat pizza while they solve problem; similarly $C$ is the subset of students who drink coffee, and $M$ is the subset of students who listen to music. Given that:
- there are $225$ students in the set $S$;
- $100$ of them eat pizza, $75$ drink coffee, and $50$ listen to music;
- $7$ students eat pizza and drink coffee, $4$ eat pizza and listen to music, while $31$ students drink coffee and listen to music;
- one student indulges in all three problem-solving stimulants, and is therefore included in all the earlier figures;
How many students are there who solve maths problems without any pizza, coffee or music?
I have calculated how many students do each of them (but only one) and from there the answer the the question as follows: $$\#(P \setminus M \setminus C) = 100 - 6 - 4 +1 = 91\\ \#(C \setminus P \setminus M) = 75 - 6 -30 +1 = 40 \\ \#(M \setminus P \setminus C) = 50 - 3 -30 +1 = 18$$ ($\#(P \setminus M \setminus C) = 100 - 6 - 4 + 1 = 91$ because there are 100 people who eat pizza, 7 of which have pizza and coffee (minus 1 that does all three), 4 that have pizza with music (minus 1 that does all three) and the one that does all).
Then, $\#(S \setminus P \setminus C \setminus M) = 225 - 90 - 40 - 18 - 6 - 3 - 30-1=35$ (#S - #(only P) - #(only C) - #(only M) - ...).
I have a feeling that this is either wrong or done in a horribly wrong way. IS it correct/is there any other way?