The coefficient of ` (x^(3)* b^(6) * C^(8) *d^(9) *e *f) ` in the expansion
of ` (a + b + c - d - e - f)^(31)` is

Options

A. 12632
B. `23110`
C. 3110
D. None of these

Explanation

To find the coefficient of \( x^3 b^6 c^8 d^9 e^1 f^1 \) in the expansion of \( (a + b + c - d - e - f)^{31} \), we will use the multinomial theorem.

### Step-by-Step Solution:

1. **Identify the total exponent and the terms**:
We have the expression \( (a + b + c - d - e - f)^{31} \). The total exponent is 31.

2. **Set up the multinomial coefficient**:
The general term in the expansion can be represented as:
\[ \frac{31!}{p_1! \, p_2! \, p_3! \, p_4! \, p_5! \, p_6!} \cdot a^{p_1} b^{p_2} c^{p_3} (-d)^{p_4} (-e)^{p_5} (-f)^{p_6} \]
where \( p_1 + p_2 + p_3 + p_4 + p_5 + p_6 = 31 \).

3. **Assign the powers to each variable**:
From the term \( x^3 b^6 c^8 d^9 e^1 f^1 \), we assign:
- \( p_1 = 3 \) (for \( x \))
- \( p_2 = 6 \) (for \( b \))
- \( p_3 = 8 \) (for \( c \))
- \( p_4 = 9 \) (for \( d \))
- \( p_5 = 1 \) (for \( e \))
- \( p_6 = 1 \) (for \( f \))

4. **Check the sum of powers**:
We need to ensure that the sum of the powers equals 31:
\[ 3 + 6 + 8 + 9 + 1 + 1 = 28 \]
Since \( 28 \neq 31 \), we need to adjust our powers.

5. **Calculate the remaining power**:
The remaining power is \( 31 - 28 = 3 \). This means we need to distribute 3 among the variables \( a \) (which corresponds to \( p_1 \)).

6. **Calculate the multinomial coefficient**:
The multinomial coefficient will be:
\[ \frac{31!}{3! \, 6! \, 8! \, 9! \, 1! \, 1!} \]

7. **Account for the negative signs**:
The terms \( -d, -e, -f \) contribute a factor of \( (-1)^{p_4 + p_5 + p_6} = (-1)^{9 + 1 + 1} = (-1)^{11} = -1 \).

8. **Final coefficient**:
Therefore, the coefficient of \( x^3 b^6 c^8 d^9 e^1 f^1 \) is:
\[ -\frac{31!}{3! \, 6! \, 8! \, 9! \, 1! \, 1!} \]

### Conclusion:
Since the coefficient is negative and the options do not include negative values, the answer is "none of these".