Statement-1: The number of distict term in the expansion of `(1+px)^(20)+(1-px)^(20)` is 42.
Statement-2: Number of term in the expansion of `(1+x)^(n)` is (n+1).

Options

A. Statement-1 is True, statement-2 is true, statement-2 is a correct explanationn for statement-1
B. Statement-1 is true, statement-2 is true, statement-2 is NOT a correct explanation for statement-1
C. Statement-1 is true, statement-2 is false
D. Statement-1 is false, statement-2 is true

Answer

To solve the problem, we need to analyze both statements provided in the question.

### Step 1: Understanding the Expansion of \((1 + px)^{20} + (1 - px)^{20}\)

We start with the expression:
\[ (1 + px)^{20} + (1 - px)^{20} \]

Using the Binomial Theorem, we can expand both terms:
\[ (1 + px)^{20} = \sum_{k=0}^{20} \binom{20}{k} (px)^k = \sum_{k=0}^{20} \binom{20}{k} p^k x^k \]
\[ (1 - px)^{20} = \sum_{k=0}^{20} \binom{20}{k} (-px)^k = \sum_{k=0}^{20} \binom{20}{k} (-1)^k p^k x^k \]

### Step 2: Combining the Two Expansions

Now, we combine the two expansions:
\[ (1 + px)^{20} + (1 - px)^{20} = \sum_{k=0}^{20} \binom{20}{k} p^k x^k + \sum_{k=0}^{20} \binom{20}{k} (-1)^k p^k x^k \]

### Step 3: Simplifying the Combined Expression

When we add these two expansions, we notice that terms with odd powers of \(x\) will cancel out because they will have opposite signs. Therefore, only the even powers of \(x\) will remain:
- For even \(k\), the terms will add up.
- For odd \(k\), the terms will cancel out.

### Step 4: Counting the Distinct Terms

The even powers of \(x\) from \(0\) to \(20\) are:
\[ 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \]
This gives us \(11\) distinct even terms.

Since we have two expansions, the total number of distinct terms in the combined expansion is:
\[ 11 \text{ (from } (1 + px)^{20}) + 11 \text{ (from } (1 - px)^{20}) = 22 \]

### Step 5: Conclusion on Statement 1

The statement claims that the number of distinct terms is \(42\), which is incorrect. The correct number of distinct terms is \(22\). Therefore, Statement 1 is **false**.

### Step 6: Analyzing Statement 2

Statement 2 states that the number of terms in the expansion of \((1 + x)^{n}\) is \(n + 1\). This is true because the expansion has terms from \(x^0\) to \(x^n\), which gives us \(n + 1\) terms.

### Final Conclusion

- Statement 1 is **false**.
- Statement 2 is **true**.

Thus, the answer is that Statement 1 is false and Statement 2 is true.