Find the coefficient of `t^8` in the expansion of `(1+2t^2-t^3)^9`.

Options

A. `1680`
B. `2140`
C. `2520`
D. `2730`

Explanation

To find the coefficient of \( t^8 \) in the expansion of \( (1 + 2t^2 - t^3)^9 \), we will use the multinomial theorem. Here’s a step-by-step solution:

### Step 1: Identify the terms
The expression can be rewritten as:
\[ (1 + 2t^2 - t^3)^9 \]
We need to find the coefficient of \( t^8 \) in this expansion.

### Step 2: Apply the multinomial theorem
According to the multinomial theorem, the expansion can be expressed as:
\[ \sum_{a+b+c=9} \frac{9!}{a!b!c!} (1)^c (2t^2)^b (-t^3)^a \]
This simplifies to:
\[ \sum_{a+b+c=9} \frac{9!}{a!b!c!} 2^b (-1)^a t^{2b + 3a} \]

### Step 3: Determine the conditions for \( t^8 \)
We need to find combinations of \( a \), \( b \), and \( c \) such that:
\[ 2b + 3a = 8 \]
and
\[ a + b + c = 9 \]

### Step 4: Solve for \( a \), \( b \), and \( c \)
We will consider two cases based on the values of \( a \):

#### Case 1: \( a = 2 \)
- From \( 2b + 3(2) = 8 \):
\[ 2b + 6 = 8 \implies 2b = 2 \implies b = 1 \]
- Now, substituting \( a = 2 \) and \( b = 1 \) into \( a + b + c = 9 \):
\[ 2 + 1 + c = 9 \implies c = 6 \]

#### Case 2: \( a = 0 \)
- From \( 2b + 3(0) = 8 \):
\[ 2b = 8 \implies b = 4 \]
- Now, substituting \( a = 0 \) and \( b = 4 \) into \( a + b + c = 9 \):
\[ 0 + 4 + c = 9 \implies c = 5 \]

### Step 5: Calculate the coefficients for both cases

#### For Case 1: \( (a, b, c) = (2, 1, 6) \)
The coefficient is given by:
\[ \frac{9!}{2!1!6!} \cdot 2^1 \cdot (-1)^2 \]
Calculating this:
\[ = \frac{9 \times 8 \times 7}{2 \times 1} \cdot 2 = 252 \cdot 2 = 504 \]

#### For Case 2: \( (a, b, c) = (0, 4, 5) \)
The coefficient is given by:
\[ \frac{9!}{0!4!5!} \cdot 2^4 \cdot (-1)^0 \]
Calculating this:
\[ = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} \cdot 16 = 126 \cdot 16 = 2016 \]

### Step 6: Sum the coefficients
The total coefficient of \( t^8 \) is:
\[ 504 + 2016 = 2520 \]

### Final Answer
Thus, the coefficient of \( t^8 \) in the expansion of \( (1 + 2t^2 - t^3)^9 \) is \( \boxed{2520} \).

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