Question

In a survey of 300 college students, it is found that 63 like brussels sprouts, 95 like broccoli, 57 like cauliflower, 26 like both Brussels sprouts and broccoli, 22 like both Brussels sprouts and cauliflower, 25 like both broccoli and cauliflower, and 13 of the students like all three vegetables.

a) How many of the 300 college students do not like any of these three vegetables?

Answer 1

To find the number of students who do not like Brussels sprouts, broccoli, or cauliflower, we use the inclusion-exclusion principle. After considering the overlaps and those who like all three, we determine that 45 out of the 300 college students do not like any of the three vegetables.

Step-by-step explanation:

Calculating the Number of Students Not Liking Vegetables

To calculate how many of the 300 college students do not like any of the three vegetables: Brussels sprouts, broccoli, and cauliflower, we utilize the principle of inclusion-exclusion formula.

First, we add up the number of students who like each vegetable:

• Brussels sprouts: 63
• Broccoli: 95
• Cauliflower: 57

Next, we subtract the number of students who like two of the vegetables:

• Both Brussels sprouts and broccoli: 26
• Both Brussels sprouts and cauliflower: 22
• Both broccoli and cauliflower: 25

Then, we add back the number of students who like all three, as they have been subtracted twice:

Students who like all three vegetables: 13

Now, we perform the calculation:

Total who like at least one vegetable = (63 + 95 + 57) - (26 + 22 + 25) + 13 = 255

To find the number of students who do not like any of the vegetables, we subtract this number from the total number of students surveyed:

Students who don't like any vegetables = Total students - Total who like at least one vegetable = 300 - 255 = 45

Therefore, 45 of the 300 college students do not like any of these three vegetables.