Students' choice problem

Question

There are 100 students who passed the university entrance exam. Of these, 55 students chose music, 44 students chose sports, and 20 students chose both. How many students chose neither music nor sports?

Penalty Shooters

Answer 1

So in this scenario, we must break down the different groups and account for the double counting of students who chose both. We then must subtract the total number of students who chose music, sports, or both from the total number of students (100) to find how many students chose neither music nor sports.

Let's use the following identity: P(A or B) = P(A) + P(B) - P(A and B)

The logic behind this equation is that it takes up all the people who chose A, all the people who chose B, and recognizes that some people chose both so you have to subtract them from the total.

In this problem, P(A) can represent students who chose music, which is 55, P(B) represents the number of students who chose 44, and P(A and B) are the students who chose both which is 20. So, we add 55+44 and subtract 20 to get 79 which is the number of students who chose either music or sports. Now, we subtract 79 from 100 (total number of students) to find the number of students who did NOT choose music nor sports, and we are left with 21.