How many committees of 5 members each can be formed with 8 officials and 4 non - official members if a particular official members is never included.

Answer

To solve the problem of forming committees of 5 members from a group of 8 officials and 4 non-officials, where one particular official is never included, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Members**:
We have 8 official members and 4 non-official members. Therefore, the total number of members is:
\[ 8 + 4 = 12 \text{ members} \]

2. **Exclude the Particular Official**:
Since one particular official is never included in the committee, we subtract that official from the total:
\[ 8 - 1 = 7 \text{ officials} \]
Thus, the total number of available members becomes:
\[ 7 + 4 = 11 \text{ members} \]

3. **Choose the Committee Size**:
We need to form a committee of 5 members from these 11 available members.

4. **Use the Combination Formula**:
The number of ways to choose \( r \) members from \( n \) members is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 11 \) and \( r = 5 \):
\[ \binom{11}{5} = \frac{11!}{5!(11-5)!} = \frac{11!}{5! \cdot 6!} \]

5. **Calculate the Factorials**:
We can simplify this expression:
\[ \binom{11}{5} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} \]

6. **Perform the Multiplication and Division**:
Calculate the numerator:
\[ 11 \times 10 = 110 \]
\[ 110 \times 9 = 990 \]
\[ 990 \times 8 = 7920 \]
\[ 7920 \times 7 = 55440 \]
Now calculate the denominator:
\[ 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
Now divide the results:
\[ \frac{55440}{120} = 462 \]

7. **Final Answer**:
Therefore, the total number of ways to form the committee is:
\[ \boxed{462} \]