Question

in a survey of 90 resorts, it was found that
28 had a spa. 39 had a childrens club. 55 had a fitness center.
11 had a spa and childrens club. 14 had a spa and a fitness center.
8 had all three features. 21 had a fitness center and childrens club.
complete parts a) through e).
a) how many of the resorts surveyed had only a spa?
(type a whole number.)
b) how many of the resorts surveyed had exactly one of these features?
(type a whole number.)
c) how many of the resorts surveyed had at least one of these features?
(type a whole number.)
d) how many of the resorts surveyed had exactly two of these features?
(type a whole number.)
e) how many of the resorts surveyed had none of these features?
(type a whole number.)

Explanation:

Step1: Label the sets

Let \(S\) be the set of resorts with a spa, \(C\) be the set of resorts with a children's club, and \(F\) be the set of resorts with a fitness center. We know \(n(S) = 28\), \(n(C)=39\), \(n(F) = 55\), \(n(S\cap C)=11\), \(n(S\cap F)=14\), \(n(C\cap F)=21\), and \(n(S\cap C\cap F)=8\). The total number of resorts \(N = 90\).

Step2: Use the principle of inclusion - exclusion

The formula for \(n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)\)

For part (a):
The number of resorts with only a spa:

$$n(\text{only }S)=n(S)-n(S\cap C)-n(S\cap F)+n(S\cap C\cap F)$$

$$n(\text{only }S)=28 - 11-14 + 8$$

$$n(\text{only }S)=11$$

For part (b):
The number of resorts with exactly one feature:

$$n(\text{only }S)+n(\text{only }C)+n(\text{only }F)$$

First, \(n(\text{only }C)=n(C)-n(S\cap C)-n(C\cap F)+n(S\cap C\cap F)\)

$$n(\text{only }C)=39-11 - 21+8=15$$

Second, \(n(\text{only }F)=n(F)-n(S\cap F)-n(C\cap F)+n(S\cap C\cap F)\)

$$n(\text{only }F)=55-14 - 21+8=28$$

$$n(\text{exactly one})=11 + 15+28=54$$

For part (c):
The number of resorts with at least one feature:

$$n(S\cup C\cup F)=n(S)+n(C)+n(F)-n(S\cap C)-n(S\cap F)-n(C\cap F)+n(S\cap C\cap F)$$

$$n(S\cup C\cup F)=28+39 + 55-11-14-21 + 8$$

$$n(S\cup C\cup F)=84$$

For part (d):
The number of resorts with exactly two features:

$$n(S\cap C)+n(S\cap F)+n(C\cap F)-3n(S\cap C\cap F)$$

$$n(\text{exactly two})=11+14 + 21-3\times8$$

$$n(\text{exactly two})=11+14 + 21 - 24=22$$

For part (e):
The number of resorts with none of the features:

$$N - n(S\cup C\cup F)$$

$$n(\text{none})=90-84 = 6$$

Answer:

a) \(11\)
b) \(54\)
c) \(84\)
d) \(22\)
e) \(6\)