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

Explanation

To find 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 1: Understanding the Expansion
The expression \( (x - 2y + 3z)^n \) is a trinomial expansion. The number of terms in the expansion of \( (a + b + c)^n \) is given by the formula:
\[ \text{Number of terms} = \frac{(n + r - 1)!}{n! (r - 1)!} \]
where \( r \) is the number of variables (in this case, \( r = 3 \) for \( x, -2y, 3z \)).

### Step 2: Applying the Formula
For our case, we have:
\[ \text{Number of terms} = \frac{(n + 3 - 1)!}{n! (3 - 1)!} = \frac{(n + 2)!}{n! \cdot 2!} \]
This simplifies to:
\[ \text{Number of terms} = \frac{(n + 2)(n + 1)}{2} \]

### Step 3: Setting Up the Equation
We know from the problem statement that the number of terms is 45. Therefore, we can set up the equation:
\[ \frac{(n + 2)(n + 1)}{2} = 45 \]

### Step 4: Solving the Equation
Multiplying both sides by 2 to eliminate the fraction gives:
\[ (n + 2)(n + 1) = 90 \]
Now, we can expand the left-hand side:
\[ n^2 + 3n + 2 = 90 \]
Subtracting 90 from both sides results in:
\[ n^2 + 3n - 88 = 0 \]

### Step 5: Factoring the Quadratic Equation
Next, we need to factor the quadratic equation:
\[ n^2 + 3n - 88 = 0 \]
We are looking for two numbers that multiply to \(-88\) and add to \(3\). The numbers \(11\) and \(-8\) satisfy this:
\[ (n + 11)(n - 8) = 0 \]

### Step 6: Finding the Values of \( n \)
Setting each factor to zero gives us:
1. \( n + 11 = 0 \) → \( n = -11 \) (not valid since \( n \) must be non-negative)
2. \( n - 8 = 0 \) → \( n = 8 \)

### Conclusion
Thus, the value of \( n \) is:
\[ \boxed{8} \]