To find the number of different terms in the expansion of \((1-x)^{201}(1+x+x^2)^{200}\), we can break down the problem step by step.
### Step 1: Rewrite the Expression
We start with the expression:
\[
(1-x)^{201}(1+x+x^2)^{200}
\]
We can rewrite \(1+x+x^2\) as \((1-x^3)/(1-x)\) using the formula for the sum of a geometric series. However, for our purposes, we will directly expand \(1+x+x^2\).
### Step 2: Expand Each Factor
1. **Expansion of \((1-x)^{201}\)**:
By the binomial theorem, the expansion of \((1-x)^{201}\) gives us terms of the form:
\[
\binom{201}{k} (-x)^k = \binom{201}{k} (-1)^k x^k
\]
where \(k\) ranges from \(0\) to \(201\). Thus, the powers of \(x\) in this expansion will be \(0, 1, 2, \ldots, 201\).
2. **Expansion of \((1+x+x^2)^{200}\)**:
The expression \(1+x+x^2\) can be expanded using the multinomial theorem. The general term in the expansion can be represented as:
\[
\frac{200!}{a!b!c!} (1)^a (x)^b (x^2)^c
\]
where \(a + b + c = 200\). The powers of \(x\) will be:
\[
b + 2c
\]
Given \(b + c \leq 200\), the minimum power occurs when \(c = 0\) (which gives \(b\) from \(0\) to \(200\)), and the maximum occurs when \(b = 0\) (which gives \(2c\) from \(0\) to \(400\)). Therefore, the powers of \(x\) in this expansion can range from \(0\) to \(400\).
### Step 3: Combine the Powers
Now, we need to combine the powers from both expansions:
- From \((1-x)^{201}\), we have powers \(0, 1, 2, \ldots, 201\).
- From \((1+x+x^2)^{200}\), we have powers \(0, 1, 2, \ldots, 400\) (specifically all even numbers up to \(400\) and all odd numbers up to \(399\)).
### Step 4: Count Unique Terms
1. **Terms from \((1-x)^{201}\)**: There are \(202\) unique terms (from \(0\) to \(201\)).
2. **Terms from \((1+x+x^2)^{200}\)**: The powers of \(x\) can be \(0, 1, 2, \ldots, 400\), which gives us \(401\) unique terms.
### Step 5: Total Unique Terms
Since the powers of \(x\) from both expansions do not overlap (the first expansion has terms \(0\) to \(201\) and the second has terms \(0\) to \(400\)), we can simply add the number of unique terms from both expansions:
\[
202 + 401 = 603
\]
Thus, the total number of different terms in the expansion of \((1-x)^{201}(1+x+x^2)^{200}\) is **603**.
