Let $p\_1,p\_2,...,p\_n$ are given probabilities. ($\sum\_{i=1}^n p\_i =1, p\_i \geq 0 $). Is there any distribution, which picks $k\leq n$ distinct elements from $1,2,...,n$ such that $P(i \in S) = k p\_i$ and $(i\neq j)$ $P(i \in S, j \in S) = c p\_i p\_j$, where S is our distribution realization and $c$ is some constant?



Sampling with non-uniform probabilities
Usually not. Denote $f\_i=\mathbb{1}\_{i\in S}$, then $\mathbb{E} f\_i=\mathbb{E} f\_i^2=kp\_i$, $\sum f\_i\equiv k$, $\mathbb{E} f\_if\_j=cp\_ip\_j$ if $i\ne j$. Thus
$$k^2p\_i=k\mathbb{E} f\_i=\mathbb{E} \sum\_j f\_if\_j=cp\_i(1-p\_i)+kp\_i.$$
If $p\_i>0$, this gives $c(1-p\_i)=k^2-k$, and if $p\_i\ne p\_j$ and $k>1$ this gives a contradiction.


