Relationship Between A M And G M

Since, product of positive number is unity.
.....(i)
Using A.M. GM

[From (i)]

Step 1: Understanding the Question:
The problem asks for an inequality condition for the product of sums of three distinct positive real numbers .

Step 2: Key Formula or Approach:
The Arithmetic Mean-Geometric Mean (AM-GM) Inequality states that for positive real numbers, .
Equality holds if and only if . Since are distinct, the sum of any two will be strictly greater than twice the square root of their product.

Step 3: Detailed Explanation:
Applying AM-GM to pairs of distinct numbers :
1. For and : .
2. For and : .
3. For and : .
Multiply these three inequalities together (valid since all terms are positive):


Step 4: Final Answer:
The expression is greater than .