A system consists of n identical particles each of mass m. The total number of interaction potential energy terms possible are `(n(n-1))/x`. Find value of x.

Explanation

To find the value of \( x \) in the expression for the total number of interaction potential energy terms possible for \( n \) identical particles, we can follow these steps:

### Step 1: Understand the Problem
We have \( n \) identical particles, and we want to determine how many unique pairs of interactions can occur between these particles.

### Step 2: Identify the Interaction Pairs
Each interaction potential energy term corresponds to a unique pair of particles. For example, if we have particles labeled as \( P_1, P_2, P_3, \ldots, P_n \), the interactions can be represented as pairs like \( (P_1, P_2), (P_1, P_3), \ldots, (P_{n-1}, P_n) \).

### Step 3: Calculate the Number of Unique Pairs
The number of ways to choose 2 particles from \( n \) particles is given by the combination formula \( nC2 \), which is calculated as follows:

\[ nC2 = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2} \]

### Step 4: Relate to the Given Expression
According to the problem, the total number of interaction potential energy terms is given as \( \frac{n(n-1)}{x} \).

### Step 5: Set Up the Equation
From our calculation, we have:

\[ \frac{n(n-1)}{2} = \frac{n(n-1)}{x} \]

### Step 6: Solve for \( x \)
To find \( x \), we can equate the two expressions:

\[ \frac{n(n-1)}{2} = \frac{n(n-1)}{x} \]

By cross-multiplying, we get:

\[ n(n-1) \cdot x = n(n-1) \cdot 2 \]

Assuming \( n(n-1) \) is not zero (which is valid for \( n \geq 2 \)), we can divide both sides by \( n(n-1) \):

\[ x = 2 \]

### Conclusion
Thus, the value of \( x \) is \( 2 \).

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