Question

100 students of a school have been grouped into 3 clubs, namely the cultural club, dance club, and music club, as in the table below.

If 17 students are selected for all the three clubs, find the maximum number of students in both cultural and dance clubs but not in the music club.

• A. 17
• B. 2
• C. 26
• D. 38

Answer 1

The correct option is C 26

Let's make a Venn diagram for the situation, and mark the variables for the missing sections.

17 students who are selected for all the three clubs are represented by the intersection of all the three clubs.

By the given conditions, we see:

Cultural club:
Total students selected for the club = 60

So,
a1 + b1 + b2 + 17 = 60
⇒ a1 + b1 + b2 = 60 – 17 = 43 ...... (eq 1)

Dance club:
Total students selected for the club = 45

So,
a2 + b1 + b3 + 17 = 45
⇒ a2 + b1 + b3 = 45 – 17 = 28 .... (eq 2)

Music club:
Total students selected for the club = 55

So,
a3 + b2 + b3 + 17 = 55
⇒ a3 + b2 + b3 = 55 – 17 = 38 ..... (eq 3)

Adding all the three equations, we get:
a1 + a2 + a3 + 2(b1 + b2 + b3) = 109
⇒ a1 + a2 + a3 = 109 – 2(b1 + b2 + b3)

By the initial condition,
Total number of students = 100

So,
a1 + a2 + a3 + b1 + b2 + b3 + 17 = 100
⇒ a1 + a2 + a3 + b1 + b2 + b3 = 83


Let's substitute the value of a1 + a2 + a3:
109 – 2(b1 + b2 + b3) + (b1 + b2 + b3) = 83
⇒ b1 + b2 + b3 = 109 – 83 = 26

To maximize b1 (students selected only for cultural club and dance club), we have to minimize b2 and b3.

The minimum value of both b2 and b3 can be 0.

Hence,
b1 + b2 + b3 = 26
⇒ b1 + 0 + 0 = 26
⇒ b1 = 26

Hence, there can be a maximum of 26 students who can be selected only for the cultural and dance clubs.

The same can be seen on the Venn diagram: