In how many ways can 6 persons be selected from 4 officers and 8 constables, if atleast one officer has to be included?

Options

A. 224
B. 672
C. 896
D. None

Answer

To solve the problem of selecting 6 persons from 4 officers and 8 constables, with the condition that at least one officer must be included, we can break down the problem into several cases based on the number of officers selected.

### Step-by-Step Solution:

1. **Identify the Cases**:
We need to consider the different cases based on how many officers we select. The possible cases are:
- Case 1: Select 1 officer and 5 constables.
- Case 2: Select 2 officers and 4 constables.
- Case 3: Select 3 officers and 3 constables.
- Case 4: Select 4 officers and 2 constables.

2. **Calculate Each Case**:
We will use the combination formula \( nCr = \frac{n!}{r!(n-r)!} \) to calculate the number of ways to select officers and constables for each case.

- **Case 1**: Select 1 officer and 5 constables.
\[ \text{Ways} = \binom{4}{1} \times \binom{8}{5} = 4 \times 56 = 224 \]

- **Case 2**: Select 2 officers and 4 constables.
\[ \text{Ways} = \binom{4}{2} \times \binom{8}{4} = 6 \times 70 = 420 \]

- **Case 3**: Select 3 officers and 3 constables.
\[ \text{Ways} = \binom{4}{3} \times \binom{8}{3} = 4 \times 56 = 224 \]

- **Case 4**: Select 4 officers and 2 constables.
\[ \text{Ways} = \binom{4}{4} \times \binom{8}{2} = 1 \times 28 = 28 \]

3. **Sum Up All Cases**:
Now, we add the number of ways from all the cases to get the total number of ways to select the persons.
\[ \text{Total Ways} = 224 + 420 + 224 + 28 = 896 \]

### Final Answer:
The total number of ways to select 6 persons from 4 officers and 8 constables, ensuring at least one officer is included, is **896**.

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