To fill a number of vacancies, an employer must hire 3 programmers from among 6 applicants, and 2 managers from among 4 applicants. What is the total number of ways in which she can make her selection?

Options

A. 1490
B. 132
C. 120
D. 60

Answer

To solve the problem of hiring 3 programmers from 6 applicants and 2 managers from 4 applicants, we can use the concept of combinations. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the number of ways to choose 3 programmers from 6 applicants.
We can use the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of applicants, and \( r \) is the number of applicants to choose.

For the programmers:
- \( n = 6 \) (total programmers)
- \( r = 3 \) (programmers to be hired)

Using the formula:
\[ 6C3 = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!} \]

Calculating this:
\[ = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{120}{6} = 20 \]

### Step 2: Calculate the number of ways to choose 2 managers from 4 applicants.
Again, we use the combination formula:
For the managers:
- \( n = 4 \) (total managers)
- \( r = 2 \) (managers to be hired)

Using the formula:
\[ 4C2 = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} \]

Calculating this:
\[ = \frac{4 \times 3}{2 \times 1} = \frac{12}{2} = 6 \]

### Step 3: Calculate the total number of ways to make the selection.
To find the total number of ways to hire both programmers and managers, we multiply the number of ways to choose programmers by the number of ways to choose managers:
\[ \text{Total Ways} = (6C3) \times (4C2) = 20 \times 6 = 120 \]

### Final Answer:
The total number of ways in which the employer can make her selection is **120**.

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