The coefficient of `x^3` in the expansion of `(1-x+x^2)^5` is (a). -83 (b). 0 (c)`.^(30)C_(10)` (d). none of these

Options

A. `-83`
B. `0`
C. `.^(30)C_(10)`
D. none of these

Explanation

To find the coefficient of \( x^3 \) in the expansion of \( (1 - x + x^2)^5 \), we can use the multinomial expansion theorem. Here’s a step-by-step solution:

### Step 1: Identify the terms in the expression
The expression we need to expand is \( (1 - x + x^2)^5 \). We can treat this as a multinomial expansion where we have three terms: \( A = 1 \), \( B = -x \), and \( C = x^2 \).

### Step 2: Use the multinomial expansion formula
The multinomial expansion of \( (A + B + C)^n \) is given by:
\[ \sum_{i+j+k=n} \frac{n!}{i! j! k!} A^i B^j C^k \]
where \( i, j, k \) are non-negative integers such that \( i + j + k = n \).

### Step 3: Set up the equation for our case
In our case, we have:
\[ n = 5, \quad A = 1, \quad B = -x, \quad C = x^2 \]
We need to find the combinations of \( i, j, k \) such that \( j + 2k = 3 \) (since we want the coefficient of \( x^3 \)) and \( i + j + k = 5 \).

### Step 4: Solve for \( i, j, k \)
From \( j + 2k = 3 \):
- If \( k = 0 \), then \( j = 3 \) and \( i = 5 - 3 - 0 = 2 \) (i.e., \( (i, j, k) = (2, 3, 0) \)).
- If \( k = 1 \), then \( j = 1 \) and \( i = 5 - 1 - 1 = 3 \) (i.e., \( (i, j, k) = (3, 1, 1) \)).
- If \( k = 2 \), then \( j = -1 \) which is not valid.

Thus, the valid combinations are:
1. \( (2, 3, 0) \)
2. \( (3, 1, 1) \)

### Step 5: Calculate the coefficients for each combination
1. For \( (2, 3, 0) \):
\[ \text{Coefficient} = \frac{5!}{2!3!0!} (1)^2 (-x)^3 (x^2)^0 = \frac{120}{2 \cdot 6} \cdot (-1) = -10 \]

2. For \( (3, 1, 1) \):
\[ \text{Coefficient} = \frac{5!}{3!1!1!} (1)^3 (-x)^1 (x^2)^1 = \frac{120}{6 \cdot 1 \cdot 1} \cdot (-1) = -20 \]

### Step 6: Combine the coefficients
Now, we combine the coefficients from both valid combinations:
\[ \text{Total coefficient of } x^3 = -10 + (-20) = -30 \]

### Conclusion
Thus, the coefficient of \( x^3 \) in the expansion of \( (1 - x + x^2)^5 \) is \( -30 \).

### Final Answer
The correct option is (d) none of these.

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