The coefficient of `x^4` in the expansion of `(1+x-2x^2)^7` is

Options

A. `-81`
B. `-91`
C. 81
D. 91

Answer

To find the coefficient of \( x^4 \) in the expansion of \( (1 + x - 2x^2)^7 \), we can use the multinomial expansion. Here’s a step-by-step solution:

### Step 1: Identify the Multinomial Expansion
The expression can be expanded using the multinomial theorem. The general term in the expansion of \( (a + b + c)^n \) is given by:

\[ T = \frac{n!}{p!q!r!} a^p b^q c^r \]

where \( p + q + r = n \).

In our case, we have:
- \( a = 1 \)
- \( b = x \)
- \( c = -2x^2 \)
- \( n = 7 \)

### Step 2: Set Up the Equation
We need to find the coefficient of \( x^4 \). The term \( x^4 \) can be formed from the powers of \( b \) and \( c \):
- Let \( q \) be the power of \( x \) (from \( b \)).
- Let \( r \) be the power of \( -2x^2 \) (from \( c \)).

The contribution to the power of \( x \) from \( c \) is \( 2r \) (since \( (-2x^2)^r \) contributes \( x^{2r} \)). Thus, we have:

\[ q + 2r = 4 \]

### Step 3: Use the Condition \( p + q + r = n \)
From the multinomial expansion, we also have:

\[ p + q + r = 7 \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( q + 2r = 4 \) (Equation 1)
2. \( p + q + r = 7 \) (Equation 2)

From Equation 2, we can express \( p \) in terms of \( q \) and \( r \):

\[ p = 7 - q - r \]

### Step 5: Substitute and Solve
Substituting \( q \) from Equation 1 into the expression for \( p \):

From Equation 1, we can express \( q \):

\[ q = 4 - 2r \]

Substituting this into the expression for \( p \):

\[ p = 7 - (4 - 2r) - r = 3 + r \]

### Step 6: Find Non-Negative Integer Solutions
Now we need \( p, q, r \) to be non-negative integers. Therefore:

1. \( r \geq 0 \)
2. \( q = 4 - 2r \geq 0 \) implies \( r \leq 2 \)

Thus, \( r \) can take values \( 0, 1, 2 \).

### Step 7: Calculate Coefficients for Each Case
1. **Case \( r = 0 \)**:
- \( q = 4 \), \( p = 3 \)
- Coefficient:
\[ \frac{7!}{3!4!0!} \cdot (-2)^0 = \frac{5040}{6 \cdot 24} = 35 \]

2. **Case \( r = 1 \)**:
- \( q = 2 \), \( p = 4 \)
- Coefficient:
\[ \frac{7!}{4!2!1!} \cdot (-2)^1 = \frac{5040}{24 \cdot 2} \cdot (-2) = -210 \]

3. **Case \( r = 2 \)**:
- \( q = 0 \), \( p = 5 \)
- Coefficient:
\[ \frac{7!}{5!0!2!} \cdot (-2)^2 = \frac{5040}{120 \cdot 2} \cdot 4 = 84 \]

### Step 8: Sum the Coefficients
Now, we sum the coefficients from all cases:

\[ 35 - 210 + 84 = -91 \]

### Final Answer
Thus, the coefficient of \( x^4 \) in the expansion of \( (1 + x - 2x^2)^7 \) is:

\[ \boxed{-91} \]