Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms:
$$\sum_{i<j}|x_i -x_j |.$$
1. **Understanding the Problem:**
   We are given \( n \) real numbers \( x_1, x_2, \ldots, x_n \) in the interval \([0,1]\). We need to determine the maximum value of the sum of the \(\frac{n(n-1)}{2}\) terms:
   \[
   \sum_{i<j} |x_i - x_j|.
   \]

2. **Analyzing the Objective Function:**
   The function \( |x_i - x_j| \) is convex for each pair \((x_i, x_j)\). This implies that the maximum value of the sum of these terms occurs at the boundary points of the interval \([0,1]\). Therefore, each \( x_i \) should be either 0 or 1 to maximize the sum.

3. **Considering the Boundary Points:**
   Suppose there are \( k \) zeros and \( n-k \) ones among the \( n \) numbers. We need to calculate the sum:
   \[
   \sum_{i<j} |x_i - x_j|.
   \]
   If \( x_i = 0 \) and \( x_j = 1 \), then \( |x_i - x_j| = 1 \). The number of such pairs is \( k(n-k) \), because each zero can pair with each one.

4. **Maximizing the Sum:**
   The sum \( k(n-k) \) is a quadratic function in \( k \). To find the maximum value, we can complete the square or use calculus. The function \( k(n-k) \) is maximized when \( k = \frac{n}{2} \). 

   - If \( n \) is even, \( k = \frac{n}{2} \).
   - If \( n \) is odd, \( k \) can be either \( \left\lfloor \frac{n}{2} \right\rfloor \) or \( \left\lceil \frac{n}{2} \right\rceil \).

5. **Calculating the Maximum Value:**
   - For even \( n \):
     \[
     k = \frac{n}{2}, \quad \text{so} \quad k(n-k) = \left(\frac{n}{2}\right)\left(\frac{n}{2}\right) = \frac{n^2}{4}.
     \]
   - For odd \( n \):
     \[
     k = \left\lfloor \frac{n}{2} \right\rfloor \quad \text{or} \quad k = \left\lceil \frac{n}{2} \right\rceil.
     \]
     Both values of \( k \) will give:
     \[
     k(n-k) = \left\lfloor \frac{n}{2} \right\rfloor \left( n - \left\lfloor \frac{n}{2} \right\rfloor \right) = \left\lceil \frac{n}{2} \right\rceil \left( n - \left\lceil \frac{n}{2} \right\rceil \right).
     \]
     This value is also approximately \(\frac{n^2}{4}\).

6. **Conclusion:**
   The maximum value of the sum \(\sum_{i<j} |x_i - x_j|\) is \(\left\lfloor \frac{n^2}{4} \right\rfloor\).

The final answer is \( \boxed{ \left\lfloor \frac{n^2}{4} \right\rfloor } \).