Question

on a survey of 100 boys it was found that 50 used white shirt 40 red and 30 blue. 20 used red and white , 15 red and blue and 10 blue and white. find the no of boys using all three shirts​

Answer 1

Let W be the set of boys using white shirts

Let R be the set of boys using Red shirts.

Let B be the the set of boys using Blue shirts.

GIVEN

n(W) = 50

n(R) = 40

n(B) = 30

n(RUW) = 20

n(RUB) = 15

n(WUB) = 10

n(WUBUR) = 100

TO FIND

n(R∩W∩B) = ?

SOLUTION

By Formula, we know that

n(WURUB)

n(WURUB) =n(W) + n(R) + n(B) - n(WUR) - n(WUB) - n(RUB) + n(R∩W∩B)

Substituting values,

100 = 50 + 40 + 30 - 20 - 15 - 10 + n(R∩W∩B)

=> 100 = 120 - 45 + n(R∩W∩B)

=> n(R∩W∩B) = 100 - 120 + 45

=> n(R∩W∩B) = 145 - 120

=> n(R∩W∩B) = 25

Hence the number of boys using all the three shirts is 25. (ANS)