Answer to Question #278750 in Statistics and Probability for feey

Question

In a survey of meals taken by 104 students during lunch, the following data were obtained: 88 students eat at least Sandwich, Salad or Salmon; 45 students eat Sandwich; 37 students eat Salad; 45 students eat Salmon; 13 students eat Sandwich and Salad; 13 students eat Salad and Salmon ; 20 students eat Sandwich and Salmon. Based on the survey, calculate the number of students who:

1. do not eat any of Sandwich, Salad or Salmon for lunch?

2. eat all Sandwich, Salad and Salmon for lunch?

3. eat Salad or Salmon?

4. do not eat Sandwich for lunch?

5. do not eat Sandwich and Salad for lunch?

6. do not eat Sandwich or Salad for lunch?

7. eat Salmon but do not eat Sandwich or Salad for lunch?

8. only eat one meal either Sandwich only, or Salad only or Salmon only for lunch?r

9. do not eat Salmon but eat Sandwich and Salad for lunch?

10. eat at least two meals for lunch?

Answer 1

The information above can be represented in the Venn diagram below.

$1)$

The number of students who do not eat any of Sandwich, Salad or Salmon for lunch is 104-88=16.

$2)$

We are given the following,

Number of students who take sandwich or salad or salmon is 88

Number of students who take sandwich is 45

Number of students who take salmon is 45

Number of students who take salad is 37

Number of students who take sandwich and salad is 13

Number of students who take salad and salmon is 13

Number of students who take sandwich and salmon is 20

We find the number of students who eat all sandwich, salad and salmon as follows.

(sandwich or salad or salmon)=(sandwich)+(salmon)+(salad)-(sandwich and salad)-(sandwich and salmon)-(salmon and salad)+(sandwich and salad and salmon)

Therefore we have,

88=45+45+37-13-13-20+(sandwich and salad and salmon).

Thus,

sandwich and salad and salmon=88-81=7

Therefore, the number of employees who eat all Sandwich, Salad and Salmon for lunch is 7.

$3)$

The number of students who eat Salad or Salmon for lunch is 19+13+7+6+6+18=69

$4)$

The number of students who do not eat Sandwich for lunch is 19+6+18=43

$5)$

The number of students who do not eat Sandwich and Salad for lunch 19+13+6+19+18=75.

$6)$

The number of students who do not eat Sandwich or Salad for lunch is 19.

$7)$

The number of students who eat Salmon but do not eat Sandwich or Salad for lunch is 19.

$8)$

The number of students who only eat one meal either Sandwich only, or Salad only or Salmon only for lunch is 19+19+18=56.

$9)$

The number of students who do not eat Salmon but eat Sandwich and Salad for lunch is 6.

$10)$

The number of students who  eat at least two meals for lunch is 13+7+6+6=32.