Coefficient of `a^8b^6c^4` in `(a+b+c)^18` is

Options

A. `(18!)/(4!10!5!)`
B. `(18!)/(3!8!8!)`
C. `(18!)/(2!7!9!)`
D. `(18!)/(8!6!4!)`

Answer

To find the coefficient of \( a^8 b^6 c^4 \) in the expansion of \( (a + b + c)^{18} \), we can use the multinomial theorem. The multinomial theorem states that:

\[ (x_1 + x_2 + \ldots + x_m)^n = \sum_{k_1 + k_2 + \ldots + k_m = n} \frac{n!}{k_1! k_2! \ldots k_m!} x_1^{k_1} x_2^{k_2} \ldots x_m^{k_m} \]

In our case, we have \( x_1 = a \), \( x_2 = b \), \( x_3 = c \), and \( n = 18 \). We need to find the coefficient of the term \( a^8 b^6 c^4 \).

### Step 1: Identify the powers
We need to identify the powers of \( a \), \( b \), and \( c \):
- The power of \( a \) is 8.
- The power of \( b \) is 6.
- The power of \( c \) is 4.

### Step 2: Check the total power
We need to ensure that the sum of the powers equals the total power of the expansion:
\[ 8 + 6 + 4 = 18 \]
This is correct since it matches the total power of the expansion.

### Step 3: Use the multinomial coefficient
The coefficient of \( a^8 b^6 c^4 \) in the expansion can be calculated using the multinomial coefficient:
\[ \text{Coefficient} = \frac{18!}{8! \, 6! \, 4!} \]

### Step 4: Calculate the factorials
Now we can calculate the factorials:
- \( 18! \) is the factorial of 18.
- \( 8! \) is the factorial of 8.
- \( 6! \) is the factorial of 6.
- \( 4! \) is the factorial of 4.

### Step 5: Substitute the values
Substituting the values into the coefficient formula:
\[ \text{Coefficient} = \frac{18!}{8! \, 6! \, 4!} \]

### Step 6: Simplify the expression
You can simplify this expression by calculating the factorials:
- \( 8! = 40320 \)
- \( 6! = 720 \)
- \( 4! = 24 \)
- \( 18! = 6402373705728000 \)

Now, substituting these values:
\[ \text{Coefficient} = \frac{6402373705728000}{40320 \times 720 \times 24} \]

Calculating \( 40320 \times 720 \times 24 \):
\[ 40320 \times 720 = 29030400 \]
\[ 29030400 \times 24 = 696729600 \]

Now, dividing:
\[ \text{Coefficient} = \frac{6402373705728000}{696729600} = 9189180 \]

### Final Answer
Thus, the coefficient of \( a^8 b^6 c^4 \) in \( (a + b + c)^{18} \) is \( 9189180 \).

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