In the expansion of `(x +y + z)^25`

Options

A. every term is of the term form `.^(25)C_(r), .^(r )C_(k).x^(25-r), y^(r-k). z^(k)`
B. the coefficient of `x^(8)y^(9)z^(9)` is 0
C. the number of terms is 325
D. None of these

Answer

To solve the problem of finding the coefficients in the expansion of \((x + y + z)^{25}\), we can apply the multinomial theorem. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Multinomial Theorem
The multinomial theorem states that the expansion of \((x_1 + x_2 + ... + x_m)^n\) can be expressed as:
\[ \sum_{k_1 + k_2 + ... + k_m = n} \frac{n!}{k_1! k_2! ... k_m!} x_1^{k_1} x_2^{k_2} ... x_m^{k_m} \]
where the sum is taken over all non-negative integers \(k_1, k_2, ..., k_m\) such that \(k_1 + k_2 + ... + k_m = n\).

### Step 2: Identify the General Term
For our case, with \(x + y + z\) raised to the power of 25, the general term in the expansion can be expressed as:
\[ \frac{25!}{k_1! k_2! k_3!} x^{k_1} y^{k_2} z^{k_3} \]
where \(k_1 + k_2 + k_3 = 25\).

### Step 3: Determine the Coefficient of Specific Terms
We are interested in the coefficients of specific terms. For example, if we want to find the coefficient of \(x^5 y^9 z^{11}\), we set \(k_1 = 5\), \(k_2 = 9\), and \(k_3 = 11\).

### Step 4: Check the Sum of Powers
We need to ensure that the sum of the powers equals 25:
\[ k_1 + k_2 + k_3 = 5 + 9 + 11 = 25 \]
This is valid, so we can proceed to find the coefficient.

### Step 5: Calculate the Coefficient
Using the multinomial coefficient formula:
\[ \text{Coefficient} = \frac{25!}{5! \cdot 9! \cdot 11!} \]

### Step 6: Analyze Other Terms
If we consider terms like \(x^8 y^9 z^9\), we check:
\[ k_1 + k_2 + k_3 = 8 + 9 + 9 = 26 \]
Since 26 is greater than 25, the coefficient for this term is 0.

### Conclusion
From our analysis, we can conclude:
- The coefficient of \(x^5 y^9 z^{11}\) is a non-zero value calculated from the multinomial coefficient.
- The coefficient of \(x^8 y^9 z^9\) is 0 because the sum of the powers exceeds 25.