Question

in a survey of employees at a fast - food restaurant, it was determined that 10 cooked food, 6 washed dishes, 21 operated the cash register, 2 cooked food and washed dishes, 5 cooked food and operated the cash register, 3 washed dishes and operated the cash register, 1 did all three jobs, and 6 did none of these jobs. complete parts a) through f) below.
a) how many employees were surveyed?
b) how many of the employees only cooked food? (simplify your answer.) (simplify your answer.)

Explanation:

Step1: Use the principle of inclusion - exclusion

Let \(C\) be the set of employees who cooked food, \(D\) be the set of employees who washed dishes and \(R\) be the set of employees who operated the cash - register.
We know that the total number of employees \(N\) (including those who did none of the jobs) is related to \(|C|\), \(|D|\), \(|R|\), \(|C\cap D|\), \(|C\cap R|\), \(|D\cap R|\), \(|C\cap D\cap R|\) by the formula \(N = |C|+|D|+|R|-|C\cap D|-|C\cap R|-|D\cap R|+|C\cap D\cap R|+ \text{number of employees who did none of the jobs}\).
We are given: \(|C| = 10\), \(|D| = 6\), \(|R| = 21\), \(|C\cap D| = 2\), \(|C\cap R| = 5\), \(|D\cap R| = 3\), \(|C\cap D\cap R| = 1\) and the number of employees who did none of the jobs \(=6\).

Step2: Calculate the number of employees surveyed (part a)

We want to find the number of employees surveyed. Using the inclusion - exclusion formula, the number of employees surveyed \(n\) is:

$$n=(10 + 6+21-2 - 5-3 + 1)+6$$

$$n=(37 - 10+1)+6$$

$$n=(27 + 1)+6$$

$$n = 34$$

Step3: Calculate the number of employees who only cooked food (part b)

The number of employees who only cooked food is \(|C|-(|C\cap D|+|C\cap R|)+|C\cap D\cap R|\).

$$|C|-(|C\cap D|+|C\cap R|)+|C\cap D\cap R|=10-(2 + 5)+1$$

$$=10 - 7+1$$

$$=4$$

Answer:

a) 34
b) 4