Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.
To find the smallest positive real \( k \) such that for any four different real numbers \( a, b, c, d \) which are not less than \( k \), there exists a permutation \((p, q, r, s)\) of \((a, b, c, d)\) such that the equation \((x^2 + px + q)(x^2 + rx + s) = 0\) has four different real roots, we need to ensure that both quadratic equations have distinct real roots.

1. **Discriminant Conditions**:
   For the quadratic equations \(x^2 + px + q = 0\) and \(x^2 + rx + s = 0\) to have real roots, their discriminants must be positive:
   \[
   p^2 - 4q > 0 \quad \text{and} \quad r^2 - 4s > 0
   \]

2. **Ordering the Values**:
   Without loss of generality, assume \(a \leq b \leq c \leq d\). We need to find a permutation \((p, q, r, s)\) of \((a, b, c, d)\) such that the discriminants are positive and the roots are distinct.

3. **Choosing \(p, q, r, s\)**:
   Let \(p\) and \(r\) be the two larger values and \(q\) and \(s\) be the two smaller values. This ensures that \(p\) and \(r\) are large enough to make the discriminants positive. Specifically, let:
   \[
   p = c, \quad q = a, \quad r = d, \quad s = b
   \]

4. **Ensuring Distinct Roots**:
   We need to ensure that the roots of the quadratics are distinct. If there is a repeated root, we can switch \(q\) and \(s\) to try to eliminate the repeated root. However, we need to show that this arrangement guarantees distinct roots.

5. **Minimum Value of \(k\)**:
   To ensure that the discriminants are positive, we need:
   \[
   c^2 - 4a > 0 \quad \text{and} \quad d^2 - 4b > 0
   \]
   Since \(a, b, c, d \geq k\), we need:
   \[
   k^2 - 4k > 0
   \]
   Solving this inequality:
   \[
   k(k - 4) > 0
   \]
   This inequality holds for \(k > 4\). Therefore, the smallest positive \(k\) that satisfies this condition is \(k = 4\).

The final answer is \(\boxed{4}\).