Musical High School has a band that contains four members: (A)vi, (B)rianna, (C)alvin, and (D)achen. Students in the high school must vote for who they like in the band, and they can vote for none of them or for one or more members of the band. There are 3000 students in the school (including the band members), but the band members cannot vote. 1000 students vote for member A. 1000 students vote for member B. 1000 students vote for member C and 1000 vote for member D. 250 students vote for both A and B. 250 students vote for both A and B. 250 students vote for both A and C, 250 students vote for both A and D. 250 students vote for both B and C. 250 students vote for both B and D. 250 students vote for both C and D. 100 students vote for A,B, and C. 100 students vote for A,B, and D. 100 students vote for A,C, and D. 100 students vote for B,C and D. 50 students vote for all of them. What is the number of students that voted for no one?

With this situation, start from the vote for all 4, then move onto the vote of 3 out of 4, then the vote of 2 of the 4 and lastly deal with with the vote of 1 out of the 4;

I've tabulated it in the attached excel document;

First, there are 50 votes for A, B, C and D;

Then there are 100 votes for B, C and D;

Now, the 50 who voted for all 4 are included in the 100 for B, C and D because someone who votes for A, B, C and D has voted for B, C and D;

This means we need to deduct 50 from 100 to get 50, which is the number of voters who voted for B, C and D, but not A;

This is done for all the options where students voted for 3 of the 4;

For those who voted C and D, there are 250, but this will include the 50 who voted A, C and D, the 50 who voted B, C and D and the 50 who voted A, B, C and D;

So, we have to subtract 3×50 (= 150) from 250 to get 100 as is shown in the table;

The table is quite helpful as it shows clearly who will be included in the numbers given in the question info;

We do the same for all those who vote for 2 of the 4;

For those who voted just D, we need to add the votes of all those who voted for D and anyone else, which is 500, and subtract this from the total number of votes for D, which is 1000;

And similarly, we do the same for A, B and C as well;

Now, we can sum all the votes, which gives 2850 and subtract this from the total number of voters, which is 2996 (remember we have to exclude the 4 band members as they can't vote);

This gives us:

2996 - 2850 = 146