To find the number of terms in the expansion of \((1+x)^{101}(1+x^2-x)^{100}\) in powers of \(x\), we can follow these steps:
### Step 1: Analyze the first part of the expression
The first part of the expression is \((1+x)^{101}\). According to the Binomial Theorem, the number of terms in the expansion of \((1+x)^n\) is given by \(n+1\).
For \((1+x)^{101}\):
- Number of terms = \(101 + 1 = 102\)
### Step 2: Analyze the second part of the expression
The second part of the expression is \((1+x^2-x)^{100}\). To find the number of distinct terms in this expansion, we first need to identify the distinct terms in the polynomial \(1 + x^2 - x\).
The polynomial can be rewritten as:
\[
1 - x + x^2
\]
This polynomial has three distinct terms: \(1\), \(-x\), and \(x^2\).
### Step 3: Use the multinomial expansion
Using the multinomial expansion, the number of distinct terms in the expansion of \((a + b + c)^n\) is given by the number of non-negative integer solutions to the equation \(a + b + c = n\), where \(a\), \(b\), and \(c\) represent the powers of each term.
In our case, we have:
- \(a\) corresponds to \(1\)
- \(b\) corresponds to \(-x\)
- \(c\) corresponds to \(x^2\)
Thus, we need to find the number of distinct terms in the expansion of \((1 - x + x^2)^{100}\). The maximum power of \(x\) in this expansion can be calculated by considering the combinations of the powers of \(-x\) and \(x^2\).
### Step 4: Determine the maximum power of \(x\)
The maximum power of \(x\) can be computed as follows:
- If we take \(k\) times \(-x\) and \(m\) times \(x^2\), then the total power of \(x\) contributed by these terms is \(k + 2m\).
- The total number of terms taken from \(1\), \(-x\), and \(x^2\) must sum to \(100\): \(i + j + k = 100\), where \(i\) is the number of times \(1\) is chosen, \(j\) is the number of times \(-x\) is chosen, and \(k\) is the number of times \(x^2\) is chosen.
The maximum power of \(x\) occurs when we maximize \(k + 2m\) under the constraint \(i + j + k = 100\).
### Step 5: Calculate the distinct powers of \(x\)
The minimum power of \(x\) is \(0\) (when we choose \(1\) only), and the maximum power occurs when we take \(0\) from \(1\) and \(100\) from \(x^2\), which gives us:
- Maximum power = \(2 \times 100 = 200\)
### Step 6: Count the distinct powers
The distinct powers of \(x\) range from \(0\) to \(200\), which gives us:
- Total distinct powers = \(200 - 0 + 1 = 201\)
### Step 7: Combine the results
Now, we combine the results from both parts:
- From \((1+x)^{101}\), we have \(102\) terms.
- From \((1+x^2-x)^{100}\), we have \(201\) distinct terms.
### Final Calculation
The total number of distinct terms in the product \((1+x)^{101}(1+x^2-x)^{100}\) is given by the product of the number of terms from each part:
\[
\text{Total distinct terms} = 102 \times 201
\]
### Conclusion
Thus, the number of terms in the expansion of \((1+x)^{101}(1+x^2-x)^{100}\) in powers of \(x\) is \(102 \times 201\).
