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Mark each box of an $n\times n$ table with probability $q$. By inclusion-exclusion the difference RHS-LHS equals to the probability that there exists a full row (with all boxes marked) but there does not exist a full column: that's because for given $k$ rows the probability that they are full but no column is not full equals $(q^k-q^n)^n$.

Mark each box of an $n\times n$ table with probability $q$. By inclusion-exclusion the difference RHS-LHS equals to the probability that there exists a full row (with all boxes marked) but there does not exist a full column: that's because for given $k$ rows the probability that (they are full but no column is full) equals $(q^k-q^n)^n$.

Mark each box of an $n\times n$ table with probability $q$. By inclusion-exclusion the difference RHS-LHS equals to the probability that there exists a full line (with all boxes marked) but there does not exist a full column: that's because for given $k$ rows the probability that they are full but no column is not full equals $(q^k-q^n)^n$.

Mark each box of an $n\times n$ table with probability $q$. By inclusion-exclusion the difference RHS-LHS equals to the probability that there exists a full row (with all boxes marked) but there does not exist a full column: that's because for given $k$ rows the probability that they are full but no column is not full equals $(q^k-q^n)^n$.

Mark each box of an $n\times n$ table with probability $q$. By inclusion-exclusion the difference RHS-LHS equals to the probability that there exists a full line (with all boxes marked) but there does not exist a full column.

Mark each box of an $n\times n$ table with probability $q$. By inclusion-exclusion the difference RHS-LHS equals to the probability that there exists a full line (with all boxes marked) but there does not exist a full column: that's because for given $k$ rows the probability that they are full but no column is not full equals $(q^k-q^n)^n$.