Which of the following statement is true
Statement - I Coefficient of `x^3` in `e^(5x)` is `(5^2)/(3!)`
Statement - II `sum_(n=1)^(oo)(""^(n)C_0+""^nC_1+""^(n)C_2+.......+""^nC_n)/(""^(n)P_n)=e^2-1 `

Options

A. only I
B. only II
C. both I and II
D. neither I nor II

Answer

To determine which of the given statements is true, we will analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** The coefficient of \(x^3\) in \(e^{5x}\) is \(\frac{5^2}{3!}\).

1. The series expansion of \(e^x\) is given by:
\[ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \]
2. For \(e^{5x}\), we substitute \(x\) with \(5x\):
\[ e^{5x} = 1 + \frac{5x}{1!} + \frac{(5x)^2}{2!} + \frac{(5x)^3}{3!} + \ldots \]
3. The term for \(x^3\) in this expansion is:
\[ \frac{(5x)^3}{3!} = \frac{125x^3}{3!} \]
4. Therefore, the coefficient of \(x^3\) is:
\[ \frac{125}{3!} = \frac{125}{6} \]
5. Since \(5^2 = 25\), the statement claims the coefficient is \(\frac{25}{6}\), which is incorrect because the correct coefficient is \(\frac{125}{6}\).

**Conclusion for Statement I:** This statement is **false**.

### Step 2: Analyze Statement II
**Statement II:** \(\sum_{n=1}^{\infty} \left( \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \ldots + \binom{n}{n} \right) / P(n) = e^2 - 1\).

1. The numerator \(\sum_{k=0}^{n} \binom{n}{k}\) represents the sum of the binomial coefficients, which equals \(2^n\) (by the binomial theorem).
2. The denominator \(P(n)\) is defined as \(n!\), so we rewrite the expression:
\[ \sum_{n=1}^{\infty} \frac{2^n}{n!} \]
3. This series is recognized as the Taylor series expansion for \(e^x\) evaluated at \(x = 2\):
\[ e^2 = 1 + \frac{2^1}{1!} + \frac{2^2}{2!} + \frac{2^3}{3!} + \ldots \]
4. Since the series starts from \(n=1\), we need to subtract the first term (which is 1):
\[ \sum_{n=1}^{\infty} \frac{2^n}{n!} = e^2 - 1 \]

**Conclusion for Statement II:** This statement is **true**.

### Final Conclusion
- Statement I is **false**.
- Statement II is **true**.

Thus, the correct answer is that **only Statement II is true**.