How to estimate the probability of co-occurrence of the positive integers $c_i$ and $d_i$, $1 \leq i \leq t$ drawn from the uniform range $1$ to $2^k-1$, such that $\Sigma^t_{i=1} c^2_i = \Sigma^t_{i=1} c_i\times d_i$ and $\forall i, c_i\neq d_i$ hold?

$\forall i, c_i$ and $d_i$'s are drawn from the same distribution, but each pair $(c_i, d_i, \forall i \neq j)$ is drawn from different distribution.

To simplify the problem for the case $t=2$, it is the probability of co-occurrence of four positive integers $c_1, c_2, d_1$ and $d_2$ such that $c^2_1 + c^2_2 = c_1\times d_1 + c_2\times d_2 $ and $c_1\neq d_1$ and $c_2\neq d_2$ hold. The pairs $(c_1,d_1)$ and $(c_2,d_2)$ are chosen from two different distributions of the same range $1$ to $2^k-1$.