If the number of terms in the expansion of `(1+x)^(101)(1+x^(2)-x)^(100)` is n, then the value of `(n)/(25)` is euqal to

Answer

To solve the problem of finding the number of terms in the expansion of \((1+x)^{101}(1+x^2-x)^{100}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1+x)^{101}(1+x^2-x)^{100} \]
We can rewrite \((1+x^2-x)\) as \((1+x^2-x) = (1+x^2) - x\).

### Step 2: Factor the second term
Notice that:
\[ 1 + x^2 - x = 1 + x^2 + (-x) = 1 + x^2 + (-1)x \]
This can be recognized as a polynomial that can be analyzed for its roots or terms.

### Step 3: Use the binomial expansion
Using the binomial expansion, we know that:
\[ (1+x)^{101} \]
will have \(101 + 1 = 102\) terms (from \(x^0\) to \(x^{101}\)).

### Step 4: Analyze the second term
Now, we need to analyze \((1+x^2-x)^{100}\). We can rewrite it as:
\[ (1 + x^2 - x)^{100} \]
To find the number of distinct terms in this expansion, we need to consider the possible powers of \(x\) that can be formed.

### Step 5: Find the number of distinct terms
The expression \((1 + x^2 - x)\) can be treated as a combination of three terms: \(1\), \(x^2\), and \(-x\). The maximum degree of \(x\) in this expansion will be determined by the combinations of these terms.

### Step 6: Determine the maximum degree
The highest power of \(x\) that can be formed from \((1 + x^2 - x)^{100}\) will be:
- From \(x^2\), we can take \(k\) times, contributing \(2k\) to the degree.
- From \(-x\), we can take \(m\) times, contributing \(m\) to the degree.
- The remaining terms will be from \(1\).

The total degree will be \(2k + m\) where \(k + m \leq 100\). The possible values of \(2k + m\) will range from \(0\) to \(200\) (when \(k = 100\) and \(m = 0\)).

### Step 7: Count the distinct terms
The distinct powers of \(x\) will be:
- For \(k = 0\), \(m\) can be \(0\) to \(100\) giving \(0\) to \(100\).
- For \(k = 1\), \(m\) can be \(0\) to \(99\) giving \(2\) to \(101\).
- Continuing this way, we can see that the terms will cover all integers from \(0\) to \(200\).

### Step 8: Total distinct terms
Thus, the total number of distinct terms in the expansion is \(202\) (from \(x^0\) to \(x^{200}\)).

### Step 9: Calculate \(n/25\)
Now that we have \(n = 202\), we need to find:
\[ \frac{n}{25} = \frac{202}{25} \]
Calculating this gives:
\[ \frac{202}{25} = 8.08 \]

### Final Answer
Thus, the value of \(\frac{n}{25}\) is \(8.08\).