The number of ordered triplets (x, y, z), such that x, y, z are distinct prime numbers and xy + yz + zx = 120 is

To solve this problem, we need to understand the properties of prime numbers and how they interact in the given equation.

Step 1: Understand the problem We are looking for distinct prime numbers x, y, and z that satisfy the equation xy + yz + zx = 120.

Step 2: Break down the equation We can rewrite the equation as x(y+z) + yz = 120.

Step 3: Consider the properties of prime numbers The smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

Step 4: Trial and error We can start by plugging in the smallest prime numbers into the equation and see if they satisfy the equation.

Step 5: Find the solution After trying different combinations, we find that the only triplets that satisfy the equation are (5, 7, 17) and (5, 17, 7).

So, there are 2 ordered triplets (x, y, z) that satisfy the given conditions.