The sum of the coefficients in the expansion of `(1 - x + x^(2) - x^(3))^(n)` ,is

Options

A. 0
B. `2^(n)`
C. `3^(n)`
D. `4^(n)`

Answer

To find the sum of the coefficients in the expansion of \( (1 - x + x^2 - x^3)^n \), we can follow these steps:

### Step 1: Understand the expression
The expression we need to analyze is \( (1 - x + x^2 - x^3)^n \). The sum of the coefficients in a polynomial can be found by substituting \( x = 1 \) into the polynomial.

### Step 2: Substitute \( x = 1 \)
We substitute \( x = 1 \) into the expression:
\[ (1 - 1 + 1^2 - 1^3)^n \]

### Step 3: Simplify the expression
Now, we simplify the expression inside the parentheses:
\[ 1 - 1 + 1 - 1 = 0 \]

### Step 4: Raise to the power of \( n \)
Now, we take the result from the previous step and raise it to the power of \( n \):
\[ 0^n \]

### Step 5: Determine the result
Since \( 0^n = 0 \) for any positive integer \( n \), we conclude that the sum of the coefficients in the expansion is:
\[ 0 \]

### Final Answer
The sum of the coefficients in the expansion of \( (1 - x + x^2 - x^3)^n \) is \( 0 \).

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