If p,q,r be three distinct real numbers, then the value of `(p+q)(q+r)(r+p),` is

Options

A. `gt8pqr`
B. `lt8pqr`
C. `8pqr`
D. none of these

Explanation

To solve the problem, we need to analyze the expression \((p+q)(q+r)(r+p)\) given that \(p\), \(q\), and \(r\) are three distinct real numbers. We will use the Arithmetic Mean-Geometric Mean (AM-GM) inequality to derive the result.

### Step-by-Step Solution:

1. **Apply AM-GM Inequality**:
We start by applying the AM-GM inequality to the pairs of numbers \(p\) and \(q\):
\[ \frac{p + q}{2} \geq \sqrt{pq} \]
This implies:
\[ p + q \geq 2\sqrt{pq} \]

2. **Apply AM-GM to the next pair \(q\) and \(r\)**:
Similarly, for \(q\) and \(r\):
\[ \frac{q + r}{2} \geq \sqrt{qr} \]
This implies:
\[ q + r \geq 2\sqrt{qr} \]

3. **Apply AM-GM to the last pair \(r\) and \(p\)**:
For \(r\) and \(p\):
\[ \frac{r + p}{2} \geq \sqrt{rp} \]
This implies:
\[ r + p \geq 2\sqrt{rp} \]

4. **Combine the inequalities**:
Now, we can multiply the three inequalities together:
\[ (p + q)(q + r)(r + p) \geq (2\sqrt{pq})(2\sqrt{qr})(2\sqrt{rp}) \]
Simplifying the right side:
\[ (p + q)(q + r)(r + p) \geq 8\sqrt{pqr \cdot pqr} = 8pqr \]

5. **Conclusion**:
Since \(p\), \(q\), and \(r\) are distinct real numbers, the equality in AM-GM does not hold, which means:
\[ (p + q)(q + r)(r + p) > 8pqr \]
Therefore, the value of \((p + q)(q + r)(r + p)\) is greater than \(8pqr\).

### Final Answer:
\[ (p + q)(q + r)(r + p) > 8pqr \]