Question

There are 200 students in a class out of which 40% opt for dance, 30% opt for music and 50% for debate as extracurricular activities. 28 of these students do not choose any activity. What is the maximum number of students who opt for exactly one of these activities?

• A. 138
• B. 104
• C. 78
• D. none of these

Answer 1

The correct option is A 138

No. of students who opt for atleast 1 activity = 200 - 28 = 172
Now, 172 = 80 + 60 + 100 - n(D&M) - n(M&De) - n(D&De) + n(D&M&De)
$⟹$ n(D&M) + n(M&De) + n(D&De) = 68 + n(D&M&De).....(1)

Now, n(D) + n(M) + n(De) - 2$\times$[n(D&M) + n(M&De) + n(D&De)] + 3$\times$n(D&M&De) (we need to find the max. value for this)
= 240 - 2$\times$[68 + n(D&M&De)] + 3$\times$n(D&M&De) {from (1)}
= 104 + n(D&M&De).....(2)

Value of n(D&M&De) for which (2) is maximum and which also satisfies equation (1) will be such that -
n(D&M) = n(M&De) = n(D&De) = n(D&M&De) = 34
$\therefore$ Maximum number of students who opt for exactly one of these activities = 104 + 34 = 138.