JEE Main & Advanced Mathematics Sets Question Bank Self Evaluation Test - Sets

In a class of 80 students numbered a to 80, all odd numbered students opt if Cricket, students whose numbers are divisible by 5 opt for football and those whose numbers are divisible by 7 opt for Hockey. The number of students who do not opt any of the three games, is

A) 13

B) 24

C) 28

D) 52

Correct Answer: C

Solution :

[c]Numbers which are divisible by 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80 they are 16 in numbers, now, number which are divisible by 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77 they are 11 in numbers.

Also, total odd numbers = 40

Let C represents the students who opt. for cricket, F for football and h for hockey.

\[\therefore \] we have \[n(C)=40,\,\,n(F)=16,\,\,n(H)=11\] now, \[C\cap F=\] odd numbers which are divisible by 5.

\[C\cap H\]= Odd numbers which are divisible by 7.

\[F\cap H=\]Numbers which are divisible by both 5 and 7.

\[n(C\cap F),\,\,8,\,\,n(C\cap H)=6,\]

\[n(F\cap H)=2,n(C\cap F\cap H)=1\]

We know

\[n(C\cup F\cup H)=n(C)+n(F)+n(H)-n(C\cap F)\]

\[-n(C\cap H)-n(F\cap H)+n(C\cap H\cap F)\]

\[n(C\cup F\cup H)=67-16+1=52\]

\[\therefore n(C''\cap F'\cap H')\]

= Total students \[-n(C\cup F\cup H)\]

\[n(C'\cap F'\cap H')=80-52=28\]

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