From 6 programmers and 4 typists, an office wants to recruit 5 people. What is the number of ways this can be done so as to recruit at least one typist?

Options

A. 209
B. 210
C. 246
D. 242

Answer

To solve the problem of recruiting 5 people from 6 programmers and 4 typists with the condition that at least one typist must be included, we can break down the solution into several cases based on the number of typists recruited.

### Step-by-step Solution:

1. **Identify the total number of programmers and typists**:
- Programmers = 6
- Typists = 4
- Total = 6 + 4 = 10

2. **Define the cases based on the number of typists recruited**:
- Case 1: 1 typist and 4 programmers
- Case 2: 2 typists and 3 programmers
- Case 3: 3 typists and 2 programmers
- Case 4: 4 typists and 1 programmer

3. **Calculate the number of ways for each case**:

**Case 1: 1 typist and 4 programmers**
- Choose 1 typist from 4: \( C(4, 1) = 4 \)
- Choose 4 programmers from 6: \( C(6, 4) = 15 \)
- Total ways for Case 1: \( 4 \times 15 = 60 \)

**Case 2: 2 typists and 3 programmers**
- Choose 2 typists from 4: \( C(4, 2) = 6 \)
- Choose 3 programmers from 6: \( C(6, 3) = 20 \)
- Total ways for Case 2: \( 6 \times 20 = 120 \)

**Case 3: 3 typists and 2 programmers**
- Choose 3 typists from 4: \( C(4, 3) = 4 \)
- Choose 2 programmers from 6: \( C(6, 2) = 15 \)
- Total ways for Case 3: \( 4 \times 15 = 60 \)

**Case 4: 4 typists and 1 programmer**
- Choose 4 typists from 4: \( C(4, 4) = 1 \)
- Choose 1 programmer from 6: \( C(6, 1) = 6 \)
- Total ways for Case 4: \( 1 \times 6 = 6 \)

4. **Add the total ways from all cases**:
- Total ways = Case 1 + Case 2 + Case 3 + Case 4
- Total ways = \( 60 + 120 + 60 + 6 = 246 \)

### Final Answer:
The total number of ways to recruit 5 people such that at least one typist is included is **246**.

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