The coefficient of `x^(8) y^(6) z^(4)` in the expansion of `(x + y + z)^(18)`, is not equal to

Options

A. `""^(18)C_(14) xx""^(14)C_(8)`
B. `""^(18)C_(10)xx""^(10)C_(6)`
C. `""^(18)C_(6) xx""^(12)C_(8)`
D. `""^(18)C_(6)xx""^(14)C_(6)`

Answer

To find the coefficient of \( x^8 y^6 z^4 \) in the expansion of \( (x + y + z)^{18} \), we can use the multinomial theorem. The coefficient can be calculated using the formula:

\[ \text{Coefficient} = \frac{n!}{p! \, q! \, r!} \cdot a^p \cdot b^q \cdot c^r \]

where \( n \) is the total power (18 in this case), \( p \), \( q \), and \( r \) are the powers of \( x \), \( y \), and \( z \) respectively, and \( a \), \( b \), and \( c \) are the coefficients of \( x \), \( y \), and \( z \) (which are all 1 here).

### Step 1: Identify the values
- \( n = 18 \)
- \( p = 8 \) (power of \( x \))
- \( q = 6 \) (power of \( y \))
- \( r = 4 \) (power of \( z \))

### Step 2: Apply the formula
Substituting the values into the formula gives:

\[ \text{Coefficient} = \frac{18!}{8! \, 6! \, 4!} \]

### Step 3: Calculate the coefficient
Now we can compute the value of the coefficient:

1. Calculate \( 18! \)
2. Calculate \( 8! \), \( 6! \), and \( 4! \)
3. Substitute these values into the equation.

### Step 4: Check the options
We need to check which of the given options is not equal to \( \frac{18!}{8! \, 6! \, 4!} \).

#### Option A: \( 18C14 \)
\[ 18C14 = \frac{18!}{14! \, 4!} = \frac{18!}{(18-4)! \, 4!} \]
This can be rewritten as:
\[ = \frac{18!}{14! \, 4!} = \frac{18!}{8! \, 6! \, 4!} \]
This is equal to our coefficient.

#### Option B: \( 18C18 \)
\[ 18C18 = \frac{18!}{18! \, 0!} = 1 \]
This is not equal to our coefficient.

#### Option C: \( 18C16 \)
\[ 18C16 = \frac{18!}{16! \, 2!} = \frac{18!}{(18-2)! \, 2!} \]
This can be rewritten as:
\[ = \frac{18!}{16! \, 2!} = \frac{18!}{6! \, 4! \, 8!} \]
This is equal to our coefficient.

#### Option D: \( 18C12 \)
\[ 18C12 = \frac{18!}{12! \, 6!} = \frac{18!}{(18-6)! \, 6!} \]
This can be rewritten as:
\[ = \frac{18!}{12! \, 6!} = \frac{18!}{8! \, 4! \, 6!} \]
This is equal to our coefficient.

### Conclusion
The only option that is not equal to the coefficient \( \frac{18!}{8! \, 6! \, 4!} \) is **Option B**.