From 4 officers and 8 jawans a committee of 6 is to be chosen to include exactly one officier. The number of such committees is

Options

A. 160
B. 200
C. 224
D. 300

Answer

To solve the problem of forming a committee of 6 members from 4 officers and 8 jawans, with the condition that exactly one officer must be included, we can follow these steps:

### Step 1: Choose 1 Officer
Since we need to include exactly one officer in the committee, we first select 1 officer from the available 4 officers. The number of ways to choose 1 officer from 4 is given by the combination formula:

\[ \text{Number of ways to choose 1 officer} = \binom{4}{1} = 4 \]

### Step 2: Choose 5 Jawans
After selecting 1 officer, we need to select the remaining 5 members from the jawans. Since there are 8 jawans available, we need to choose 5 jawans from these 8. The number of ways to choose 5 jawans from 8 is given by:

\[ \text{Number of ways to choose 5 jawans} = \binom{8}{5} \]

Using the property of combinations, we know that:

\[ \binom{8}{5} = \binom{8}{3} \quad (\text{since } \binom{n}{r} = \binom{n}{n-r}) \]

Calculating \(\binom{8}{3}\):

\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

### Step 3: Calculate Total Committees
Now, we multiply the number of ways to choose the officer by the number of ways to choose the jawans to get the total number of committees:

\[ \text{Total number of committees} = \binom{4}{1} \times \binom{8}{5} = 4 \times 56 = 224 \]

Thus, the total number of committees that can be formed under the given conditions is **224**.

### Summary of Steps
1. Choose 1 officer from 4 officers: \(\binom{4}{1} = 4\)
2. Choose 5 jawans from 8 jawans: \(\binom{8}{5} = 56\)
3. Multiply the two results: \(4 \times 56 = 224\)