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I don't have many skills in probability theory, so I need a little help. My problem is the following:

Let $n_1,n_2,...,n_k\in[0,1,...,n]$ such as $n_1+n_2+...+n_k=n$. Which is the probability that $$n_1(n_1-1)+n_2(n_2-1)+...+n_k(n_k-1)\geq2m$$ where $0\leq 2m\leq n(n-1)$.

Thank you very much!

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    $\begingroup$ What distribution do you have on the set/multiset $\{n_1,...,n_k\}$? $\endgroup$
    – Douglas Zare
    Commented Aug 12, 2016 at 21:00
  • $\begingroup$ I have no specific distribution on the multiset ${n_1,...,n_k}$, but, if is mandatory in order to calculate the probability, you can assume one. Thank you! $\endgroup$
    – RaduB
    Commented Aug 13, 2016 at 6:20
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    $\begingroup$ Of course the distribution is necessary to calculate or estimate the probability. If you don't have a distribution in mind, then you don't have a question. $\endgroup$
    – Douglas Zare
    Commented Aug 13, 2016 at 7:51
  • $\begingroup$ Sorry for my mistake. I am programmer and I have only few knowledges about probabilities. I think that the distribution of ${n_1,...,n_k}$ is an uniform one as I generate them using a good PRNG, isn't? $\endgroup$
    – RaduB
    Commented Aug 13, 2016 at 15:05
  • $\begingroup$ If you generate each $n_i$ uniformly, they probably won't add up to $n$. Let me mention a completely different and very natural way to get a multiset of $k$ numbers adding up to $n$: Generate a random function from $\{1,2,...,n\}$ to $\{1,2,...,k\}$. Let $n_i$ be the count of the elements sent to $i$. The uniform distribution and this distribution will have quite different probabilities. $\endgroup$
    – Douglas Zare
    Commented Aug 13, 2016 at 17:59

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