Question

In a survey of 288 college students, it is found that 70 like brussels sprouts, 93 like broccoli, 57 like cauliflower, 28 like both brussels sprouts and broccoli, 21 like both brussels sprouts and cauliflower, 25 like both broccoli and cauliflower and 11 of the students like all three vegetables. How many of the 288 college students do not like any of these three vegetables?

Answer 1

There are 131 students do not like any of the three vegetables.

Explanation:

Total Students = 288

Students who like:

Brussels Sprouts(S) = 70 ⇒ n(S) = 70

Broccoli(B) = 93 ⇒ n(B) = 93

Cauliflower(C) =57 ⇒ n(C) = 57

n ( S∩B) = 28

n ( S∩C) = 21

n ( B∩C) = 25

n ( S∩B∩C) = 11

Now, to find the number of students who do not like any vegetable

n(A∪B∪C) = n(A)+n(B)+ n(C)- n(A∩B)- n(A∩c)- n(B∩C)+ n(A∩B∩C)

So, number of students who like either of the three vegetables is n(A∪B∪C).

⇒ Total students who like either of the vegetable is

n(S) + n(B) +n(C) - n(S∩B) -n(S∩C) - n(B∩C) + n(S∩B∩C)

= 70 + 93 + 57 - 28 - 21 - 25 + 11 = 231 - 74 = 157

So, the number of students who do not like any of the 3 vegetables

= Total Students - Number of students who like at least any of the vegetables.

= 288 - n(A∪B∪C) = 288 - 157 = 131

So, in total 131 students do not like any of the three vegetables.