Question

in a survey of employees at a fast food restaurant, it was determined that 11 cooked food, 13 washed dishes, 23 operated the cash register, 4 cooked food and washed dishes, 4 cooked food and operated the cash register, 8 washed dishes and operated the cash register, 2 did all three jobs, and 4 did none of these jobs. complete parts a) through f) below. a) how many employees were surveyed? (simplify your answer )

Explanation:

Step1: Apply the principle of inclusion - exclusion for three sets. 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. The formula for \( |C\cup D\cup R| \) is \( |C|+|D|+|R|-|C\cap D|-|C\cap R|-|D\cap R|+|C\cap D\cap R| \).

Given \( |C| = 11 \), \( |D| = 13 \), \( |R| = 23 \), \( |C\cap D| = 4 \), \( |C\cap R| = 4 \), \( |D\cap R| = 8 \), \( |C\cap D\cap R| = 2 \).
First, calculate \( |C|+|D|+|R|=11 + 13+23=47 \).
Then, calculate \( |C\cap D|+|C\cap R|+|D\cap R|=4 + 4+8 = 16 \).
Now, substitute into the formula: \( |C\cup D\cup R|=47-16 + 2=33 \).

Step2: Add the number of employees who did none of the jobs. The number of employees who did none of the jobs is 4. So the total number of employees surveyed is \( |C\cup D\cup R|+4=33 + 4=37 \).

Answer:

37