The number of terms in the expansion of `(1+2x+x^2)^(20)` when expanded in decreasing powers of `x` is

Options

A. 20
B. 21
C. 40
D. 41

Answer

To find the number of terms in the expansion of \((1 + 2x + x^2)^{20}\) when expanded in decreasing powers of \(x\), we can follow these steps:

### Step 1: Identify the structure of the expression
The expression can be rewritten in a more manageable form. We can recognize that \(1 + 2x + x^2\) can be treated as a polynomial.

### Step 2: Rewrite the polynomial
We can express \(1 + 2x + x^2\) as a perfect square:
\[ 1 + 2x + x^2 = (1 + x)^2 \]
Thus, we can rewrite the original expression as:
\[ (1 + 2x + x^2)^{20} = ((1 + x)^2)^{20} \]

### Step 3: Simplify the expression
Using the property of exponents, we can simplify further:
\[ ((1 + x)^2)^{20} = (1 + x)^{40} \]

### Step 4: Determine the number of terms
The number of terms in the expansion of \((1 + x)^n\) is given by \(n + 1\). Here, \(n = 40\):
\[ \text{Number of terms} = 40 + 1 = 41 \]

### Conclusion
Thus, the number of terms in the expansion of \((1 + 2x + x^2)^{20}\) is \(41\).