BITSAT 2024 Shift 1 Paper

In a survey of 1003 families in Calcutta, we have the following information:
63 families have neither a radio nor a TV.
794 families have a radio.
187 families have a TV.
We need to calculate how many families have both a radio and a TV.
Step-by-step Solution:
Calculate the total number of families with either a radio or a TV. Using the formula for the complement set:
n(R∪T)+n(R∪T)′=‌ Total families ‌
n(R∪T)+63=1003
Solving for n(R∪T) :
n(R∪T)=1003−63=940
Apply the principle of inclusion-exclusion from set theory:
n(R∪T)=n(R)+n(T)−n(R∩T)
Substitute the known values:
940=794+187−n(R∩T)
Simplify to find n(R∩T) :
‌940=981−n(R∩T)
‌n(R∩T)=981−940=41
Therefore, 41 families have both a radio and a TV.