Question

A survey of 500 high school students was taken to determine their favorite chocolate candy. Of the 500 students surveyed, 42 like Snickers, 110 like Twix, 124 like Reese's Peanut Butter Cups, 33 like Snickers and Twix, 62 like Twix and Reese's Peanut Butter Cups, 26 like Snickers and Reese's Peanut Butter Cups, and 22 like all three kinds of chocolate candy.

1. How many students like at most 2 kinds of these chocolate candies?
a) 55
b) 155
c) 77
d) 51
e) 478
f) None of the above

Answer 1

To find the number of students who like at most two kinds of chocolate candies, we add those who like each kind, subtract the overlaps, and then subtract those who like all three, resulting in 155 students.

Step-by-step explanation:

To calculate the number of students who like at most two kinds of these chocolate candies, we start by adding the number of students who like each individual candy, then subtract the number of students who like them in pairs as well as the number of students who like all three to avoid double counting.

• Students who like Snickers: 42
• Students who like Twix: 110
• Students who like Reese's Peanut Butter Cups: 124
• Students who like Snickers and Twix: -33
• Students who like Twix and Reese's: -62
• Students who like Snickers and Reese's: -26
• Students who like all three (Snickers, Twix, and Reese's): +22 (since they were subtracted three times, we need to add them back once)

Now we calculate: 42 + 110 + 124 - 33 - 62 - 26 + 22 = 177.

However, this includes the 22 students who like all three. Since we're interested in the number who like at most two, we need to subtract the 22 who like all three: 177 - 22 = 155.

Answer 2

Step-by-step explanation:

Universal set

U = 500

The number that likes snickers

n(S) = 42

The number that like Twix.

n(T) = 110

The number that like Reeses

n(R) = 125

n(S n T) = 33

n(T n R) = 62

n(S n R) = 26

n( S n R n T) = 22

Then,

n(S n T) only = n(S n T) - n(S n R n T)

n(S n T) only =33 - 22 = 11

n(T n R) only = n(T n R) - n(S n R n T)

n(T n R) only =62 - 22 = 40

n(S n R) only = n(S n R) - n(S n R n T)

n(S n R) only =26 - 22 = 4.

Also,

n(S) only = n(S) - n(S n R) - n(S n T) only

n(S) only = 42 - 26 - 11 = 5

n(T) only = n(T) - n(T n R) - n(T n S) only

n(T) only = 110 - 62 - 11 = 37

n(R) only = n(R) - n(S n R) - n(R n T) only

n(R) only = 124 - 26 - 40 = 58

Then, to know if some student don't like any of the of chocolate,

Let know the number of students that like the chocolate candy

n(S) + n(T)only + n(T n R)only + n(R) only

42 + 37 + 40 + 58 = 177 students.

Therefore, the total students that like chocolate candy is 177, so those that does not like any of them are

n(S U R U T)' = U – n(S U R U T)

n(S U R U T) = 500 - 177 = 323.

So, the question is how many student likes at most 2 kinds of these chocolates, this means that they can like exactly 2 or less or even none.

So, this category are

n(2 most) = n(S n T)only + n(R n T)only + n(S n R)only + n(s)only + n(T)only + n(R)only + n(S U R U T)'

n(2 most) = 11 + 40 + 4 + 5 + 37 + 58 + 323

n(2 most) = 478

The correct answer is E.

Check attachment for Venn diagram