Find the number of terms in the expansion of `(1+6x+9x^2)^(23)`

Answer

To find the number of terms in the expansion of \((1 + 6x + 9x^2)^{23}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \((1 + 6x + 9x^2)^{23}\). We can recognize that \(9x^2\) can be expressed as \((3x)^2\). Thus, we can rewrite the expression as:
\[ (1 + 6x + (3x)^2)^{23} \]

### Step 2: Identify the form
Now, we can see that the expression is in the form of \(a + b + c\), where:
- \(a = 1\)
- \(b = 6x\)
- \(c = 9x^2\)

### Step 3: Use the multinomial theorem
According to the multinomial theorem, the number of distinct terms in the expansion of \((a + b + c)^n\) is given by the formula:
\[ \frac{(n + k - 1)!}{(k - 1)!n!} \]
where \(n\) is the exponent and \(k\) is the number of different terms.

In our case:
- \(n = 23\)
- \(k = 3\) (since we have three terms: \(1\), \(6x\), and \(9x^2\))

### Step 4: Calculate the number of terms
Using the formula, we find the number of distinct terms:
\[ \text{Number of terms} = \frac{(23 + 3 - 1)!}{(3 - 1)! \cdot 23!} = \frac{25!}{2! \cdot 23!} \]

### Step 5: Simplify the expression
Now we can simplify:
\[ \frac{25!}{2! \cdot 23!} = \frac{25 \times 24}{2 \times 1} = \frac{600}{2} = 300 \]

### Step 6: Conclusion
Thus, the number of distinct terms in the expansion of \((1 + 6x + 9x^2)^{23}\) is:
\[ \boxed{300} \]