# Joint distribution of column sums when row sums are fixed

Suppose I have an $m$ by $n$ table $X_{ij} \in \{0,1\}$, where in each row, $r$ randomly chosen entries are set to 1 (the rest are 0), i.e. $\sum_j X_{ij}=r$.

I know that e.g. the column sum $\sum_i X_{i1}$ has a binomial distribution $\mathrm{Bin}(m, r/n)$. But what is the joint distribution of all column sums, i.e. of $(\sum_i X_{i1},\; ...,\;\sum_i X_{in})$ ?

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Can you write out the distribution of the whole matrix? What do you notice about the $n$th column sum given the rest? – cardinal Sep 16 '12 at 19:29