At the company YouGroove , 35 employees work in the sales department and 50 employees work in the operations department. Of these employees, 15 work in both the sales and the operations departments . How many of the 110 employees at YouGroove do NOT work in either the sales or the operations departments ?

Options

A. 10
B. 15
C. 20
D. 40

Answer

To find out how many employees at YouGroove do not work in either the sales or operations departments, we can use the principle of inclusion-exclusion. Let's break down the steps:

### Step 1: Identify the given values
- Total number of employees at YouGroove = 110
- Number of employees in the sales department (S) = 35
- Number of employees in the operations department (O) = 50
- Number of employees working in both departments (B) = 15

### Step 2: Use the inclusion-exclusion principle
According to the inclusion-exclusion principle, the total number of employees working in at least one of the departments can be calculated as:
\[ \text{Total in at least one department} = S + O - B \]
Substituting the values we have:
\[ \text{Total in at least one department} = 35 + 50 - 15 \]

### Step 3: Calculate the total in at least one department
Now, we perform the calculation:
\[ 35 + 50 = 85 \]
\[ 85 - 15 = 70 \]
So, 70 employees work in either the sales or operations departments.

### Step 4: Calculate the number of employees not working in either department
To find the number of employees who do not work in either department, we subtract the number of employees working in at least one department from the total number of employees:
\[ \text{Employees not in either department} = \text{Total employees} - \text{Total in at least one department} \]
Substituting the values:
\[ \text{Employees not in either department} = 110 - 70 \]

### Step 5: Perform the final calculation
Now, we calculate:
\[ 110 - 70 = 40 \]
Thus, the number of employees who do not work in either the sales or operations departments is **40**.

### Final Answer
The number of employees at YouGroove who do NOT work in either the sales or operations departments is **40**.

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