The value of Γ(7/2)Γ(7/2) is:a.15π−−√/215𝜋/2b.15π−−√15𝜋c.15π−−√/415𝜋/4d.15π−−√/8

To solve the problem, we will follow the outlined steps.

1. Break Down the Problem

We need to compute the value of $\Gamma\left(\frac{7}{2}\right)\Gamma\left(\frac{7}{2}\right)$ and identify its equivalent in the options provided.

2. Relevant Concepts

The Gamma function has the property that $\Gamma\left(\frac{n}{2}\right)$ can be expressed in terms of the factorial for odd integers. Specifically, we have: $\Gamma\left(\frac{n}{2}\right) = \frac{(n-1)! \sqrt{\pi}}{2^{n/2-1}}$

For $n = 7$: $\Gamma\left(\frac{7}{2}\right) = \frac{(6)! \sqrt{\pi}}{2^{7/2 - 1}} = \frac{720 \sqrt{\pi}}{2^{3.5}} = \frac{720 \sqrt{\pi}}{2^{3} \cdot \sqrt{2}} = \frac{720 \sqrt{\pi}}{8\sqrt{2}} = \frac{90 \sqrt{2} \sqrt{\pi}}{1}$

3. Analysis and Detail

Next, we compute $\Gamma\left(\frac{7}{2}\right)\Gamma\left(\frac{7}{2}\right)$: $\Gamma\left(\frac{7}{2}\right)^2 = \left(90 \frac{\sqrt{2\pi}}{1}\right)^2 = 8100 \cdot 2 \cdot \pi = 16200\pi$

Using the property: $\Gamma\left(n\right)\Gamma\left(\frac{n}{2}\right) = \frac{(\Gamma(n))^2}{\Gamma(n+1)}$ Here, for $n=7$: $\Gamma(7) = 6! = 720$

4. Verify and Summarize

Calculating each option using this value helps us determine the final representations. Since $\Gamma\left(\frac{7}{2}\right) = \frac{15\sqrt{2\pi}}{4}$, we can compute and compare values.

Final Answer

The value of $\Gamma\left(\frac{7}{2}\right)\Gamma\left(\frac{7}{2}\right)$ corresponds to the final representation among the options given.

After checking each option, we find that: $\text{The correct answer is: } \frac{15\sqrt{2\pi}}{4}$ This is option c: $\frac{15\sqrt{2}}{4}$.