If number of terms in the expansion of `(x -2y +3z)^n` are 45, then n is equal to

Explanation

To solve the problem of finding the value of \( n \) such that the number of terms in the expansion of \( (x - 2y + 3z)^n \) is 45, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of variables (p):**
In the expression \( (x - 2y + 3z) \), there are 3 distinct variables: \( x \), \( y \), and \( z \). Thus, \( p = 3 \).

2. **Use the formula for the number of terms in the expansion:**
The formula for the number of terms in the expansion of \( (x_1 + x_2 + ... + x_p)^n \) is given by:
\[ \text{Number of terms} = \binom{n + p - 1}{p - 1} \]
Substituting \( p = 3 \):
\[ \text{Number of terms} = \binom{n + 3 - 1}{3 - 1} = \binom{n + 2}{2} \]

3. **Set the equation equal to 45:**
We know from the problem statement that the number of terms is 45:
\[ \binom{n + 2}{2} = 45 \]

4. **Expand the binomial coefficient:**
The binomial coefficient can be expressed as:
\[ \binom{n + 2}{2} = \frac{(n + 2)(n + 1)}{2} \]
Setting this equal to 45 gives:
\[ \frac{(n + 2)(n + 1)}{2} = 45 \]

5. **Multiply both sides by 2 to eliminate the fraction:**
\[ (n + 2)(n + 1) = 90 \]

6. **Expand the left side:**
\[ n^2 + 3n + 2 = 90 \]

7. **Rearrange to form a quadratic equation:**
\[ n^2 + 3n + 2 - 90 = 0 \implies n^2 + 3n - 88 = 0 \]

8. **Factor the quadratic equation:**
We need to find two numbers that multiply to \(-88\) and add to \(3\). The numbers \(11\) and \(-8\) work:
\[ (n + 11)(n - 8) = 0 \]

9. **Solve for \( n \):**
Setting each factor to zero gives:
\[ n + 11 = 0 \quad \text{or} \quad n - 8 = 0 \]
This results in:
\[ n = -11 \quad \text{or} \quad n = 8 \]

10. **Determine the valid solution:**
Since \( n \) must be a non-negative integer (as it represents the power in the binomial expansion), we discard \( n = -11 \). Thus, we have:
\[ n = 8 \]

### Final Answer:
The value of \( n \) is \( 8 \).