Statement-1 Greatest coefficient in the expansion of
`(1 + 3x)^(6) " is " ""^(6)C_(3) *3^(3)` .
Statement-2 Greatest coefficient in the expansion of
`(1 + x)^(2n)` is the middle term .

Options

A. Statement I is True, Statement II is True, Statement II is a correct explanation for statement I
B. Statement I is True, Statement II is True, Statement II is NOT a correct explanation for Statement I
C. Statement I is True, Statement II is False
D. Statement I is False, Statement II is True.

Answer

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

### Step 1: Analyze Statement 2
Statement 2 claims that the greatest coefficient in the expansion of \((1 + x)^{2n}\) is the middle term.

1. The binomial expansion of \((1 + x)^{2n}\) is given by:
\[ \sum_{r=0}^{2n} \binom{2n}{r} x^{r} \]
The coefficients in this expansion are \(\binom{2n}{r}\).

2. The maximum coefficient occurs at \(r = n\) if \(2n\) is even, which corresponds to the middle term. Therefore, the middle term is indeed the greatest coefficient.

**Conclusion for Statement 2**: This statement is correct.

### Step 2: Analyze Statement 1
Statement 1 claims that the greatest coefficient in the expansion of \((1 + 3x)^6\) is \(\binom{6}{3} \cdot 3^3\).

1. The binomial expansion of \((1 + 3x)^6\) is:
\[ \sum_{r=0}^{6} \binom{6}{r} (3x)^{r} = \sum_{r=0}^{6} \binom{6}{r} 3^r x^r \]
The coefficients in this expansion are \(\binom{6}{r} \cdot 3^r\).

2. To find the greatest coefficient, we need to evaluate \(\binom{6}{r} \cdot 3^r\) for \(r = 0, 1, 2, \ldots, 6\).

3. We can check the coefficients:
- For \(r = 0\): \(\binom{6}{0} \cdot 3^0 = 1\)
- For \(r = 1\): \(\binom{6}{1} \cdot 3^1 = 6 \cdot 3 = 18\)
- For \(r = 2\): \(\binom{6}{2} \cdot 3^2 = 15 \cdot 9 = 135\)
- For \(r = 3\): \(\binom{6}{3} \cdot 3^3 = 20 \cdot 27 = 540\)
- For \(r = 4\): \(\binom{6}{4} \cdot 3^4 = 15 \cdot 81 = 1215\)
- For \(r = 5\): \(\binom{6}{5} \cdot 3^5 = 6 \cdot 243 = 1458\)
- For \(r = 6\): \(\binom{6}{6} \cdot 3^6 = 1 \cdot 729 = 729\)

4. The coefficients calculated show that the maximum coefficient occurs at \(r = 5\) which is \(1458\), not at \(r = 3\) as stated in Statement 1.

**Conclusion for Statement 1**: This statement is incorrect.

### Final Conclusion
- Statement 1 is incorrect.
- Statement 2 is correct.