A. 99

B. 90

C. 81

D. 27

E. 153

Correct Answer – Choice E. 99 students attended neither of the events.

**Explanatory Answer**

Total 180 students.

At least 45% attended the prom night.

So, the number should be a minimum of 45% of 180 = 81 students.

At least 40% took part in the debate.

So, the number should be a minimum of 40% of 180 = 72 students.

We have been asked to find out the maximum number of students who attended neither the prom night nor the debate.

If set A represents those who attended the prom night and set B represents those who attended the debate,

Set (A U B) will give us the set of students who attended at least one of the two events.

And Set (A U B)’ will give the set of students who attended neither of the two events.

If we have to maximize (A U B)’, we have to minimize (A U B).

n(A U B) = n(A) + n(B) – n(A n B)

In this question, n(A U B) = 81 + 72 – n(A n B) = 153 – n(A n B)

If n(A U B) has to be minimized, we should maximize n(A n B).

The maximum value that n(A n B) can take is the smaller of n(A) and n(B).

So, max n(A n B) = n(B) = 72.

Therefore, min n(A U B) = 153 – 72 = 81.

Hence, max n(A U B)” = 180 – 81 = 99

Rahul Beria says

Alternate method: Max no of such students vl be 55 percent…. 55 percent of 180 is 99..