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

Options

A. 7
B. 8
C. 9
D. none of these

Answer

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

### Step 1: Understand the Formula for the Number of Terms
The number of terms in the expansion of \( (a + b + c)^n \) is given by the formula:
\[ \text{Number of terms} = \binom{n + r - 1}{r - 1} \]
where \( r \) is the number of different variables in the expression. In our case, \( r = 3 \) (for \( x \), \( -2y \), and \( 3z \)).

### Step 2: Substitute the Values into the Formula
Substituting \( r = 3 \) into the formula gives:
\[ \text{Number of terms} = \binom{n + 3 - 1}{3 - 1} = \binom{n + 2}{2} \]
We know that this equals 45, so we set up the equation:
\[ \binom{n + 2}{2} = 45 \]

### Step 3: Expand the Binomial Coefficient
The binomial coefficient \( \binom{n + 2}{2} \) can be expressed as:
\[ \binom{n + 2}{2} = \frac{(n + 2)(n + 1)}{2} \]
Setting this equal to 45 gives:
\[ \frac{(n + 2)(n + 1)}{2} = 45 \]

### Step 4: Eliminate the Fraction
To eliminate the fraction, multiply both sides by 2:
\[ (n + 2)(n + 1) = 90 \]

### Step 5: Expand and Rearrange the Equation
Expanding the left side:
\[ n^2 + 3n + 2 = 90 \]
Rearranging gives:
\[ n^2 + 3n + 2 - 90 = 0 \]
which simplifies to:
\[ n^2 + 3n - 88 = 0 \]

### Step 6: Solve the Quadratic Equation
Now we can solve the quadratic equation using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = 3 \), and \( c = -88 \):
\[ n = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-88)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ n = \frac{-3 \pm \sqrt{9 + 352}}{2} \]
\[ n = \frac{-3 \pm \sqrt{361}}{2} \]
\[ n = \frac{-3 \pm 19}{2} \]

### Step 7: Calculate the Possible Values of \( n \)
Calculating the two possible values:
1. \( n = \frac{16}{2} = 8 \)
2. \( n = \frac{-22}{2} = -11 \) (not valid since \( n \) must be non-negative)

Thus, the only valid solution is:
\[ n = 8 \]

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

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