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

Answer:

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