To solve this problem, we can use the principle of Inclusion-Exclusion, which helps us count the number of elements in different sets.
Let's define the following:
A = Number of students who like Brussels sprouts.
B = Number of students who like broccoli.
C = Number of students who like cauliflower.
a) To find how many of the 287 college students do not like any of these three vegetables, we will subtract the total number of students who like at least one vegetable from the total number of students:
Total students = 287
Total students who like at least one vegetable = A + B + C - (students who like both Brussels sprouts and broccoli) - (students who like both Brussels sprouts and cauliflower) - (students who like both broccoli and cauliflower) + (students who like all three vegetables)
Total students who like at least one vegetable = 62 + 97 + 57 - 29 - 23 - 24 + 12 = 152
Number of students who do not like any of the three vegetables = 287 - 152 = 135
b) To find how many like broccoli only, we need to subtract the number of students who like both broccoli and cauliflower, and those who like all three vegetables from the total number of students who like broccoli:
Number of students who like broccoli only = (students who like broccoli) - (students who like both broccoli and cauliflower) - (students who like all three vegetables)
Number of students who like broccoli only = 97 - 24 - 12 = 61
c) To find how many like broccoli AND cauliflower but not Brussels sprouts, we need to subtract the number of students who like all three vegetables from the number of students who like both broccoli and cauliflower:
Number of students who like broccoli AND cauliflower but not Brussels sprouts = (students who like both broccoli and cauliflower) - (students who like all three vegetables)
Number of students who like broccoli AND cauliflower but not Brussels sprouts = 24 - 12 = 12
d) To find how many like neither Brussels sprouts nor cauliflower, we need to subtract the total number of students who like at least one vegetable from the total number of students:
Number of students who like neither Brussels sprouts nor cauliflower = Total students - (students who like at least one vegetable)
Number of students who like neither Brussels sprouts nor cauliflower = 287 - 152 = 135
So, the answers are:
a) 135 students do not like any of the three vegetables.
b) 61 students like broccoli only.
c) 12 students like broccoli AND cauliflower but not Brussels sprouts.
d) 135 students like neither Brussels sprouts nor cauliflower.