The expansion `1+x,1+x+x^(2),1+x+x^(2)+x^(3),….1+x+x^(2)+…+x^(20)` are multipled together and the terms of the product thus obtained are arranged in increasing powers of `x` in the form of `a_(0)+a_(1)x+a_(2)x^(2)+…`, then,
Number of terms in the product

Options

A. `200`
B. `211`
C. `231`
D. none of these

Explanation

To solve the problem, we need to analyze the product of the given expansions and determine the number of distinct terms in the resulting polynomial.

### Step-by-Step Solution:

1. **Identify the Series**: The expansions given are:
- \(1 + x\)
- \(1 + x + x^2\)
- \(1 + x + x^2 + x^3\)
- ...
- \(1 + x + x^2 + \ldots + x^{20}\)

Each expansion can be expressed as:
\[ f_n(x) = 1 + x + x^2 + \ldots + x^n \]
for \(n = 0, 1, 2, \ldots, 20\).

2. **Sum of the Series**: The series can be summed up using the formula for the sum of a geometric series:
\[ f_n(x) = \frac{1 - x^{n+1}}{1 - x} \quad \text{for } x \neq 1 \]

3. **Product of the Series**: We need to find the product:
\[ P(x) = f_0(x) \cdot f_1(x) \cdot f_2(x) \cdots f_{20}(x) \]

4. **Determine the Highest Power of \(x\)**: The highest degree of \(x\) in \(P(x)\) is obtained by summing the highest powers from each individual expansion:
\[ \text{Highest degree} = 0 + 1 + 2 + \ldots + 20 \]
This is the sum of the first 20 natural numbers, which can be calculated using the formula:
\[ S = \frac{n(n + 1)}{2} \]
where \(n = 20\):
\[ S = \frac{20 \times 21}{2} = 210 \]

5. **Count the Distinct Terms**: The polynomial \(P(x)\) will have terms from \(x^0\) to \(x^{210}\). Therefore, the total number of distinct terms is:
\[ 210 + 1 = 211 \]

### Final Answer:
The total number of terms in the product is **211**.