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Let $m=2$, $r=3$, $n=4$; then the second equation says $${6\choose3}=\sum_0^3{2\choose k}{4\choose3-k}={2\choose0}{4\choose3}+{2\choose1}{4\choose2}+{2\choose2}{4\choose1}$$ Now let $m=2$, $r=1$, $n=4$; then the first equation says $${6\choose3}=\sum_0^2{2\choose k}{4\choose k+1}={2\choose0}{4\choose1}+{2\choose1}{4\choose2}+{2\choose2}{4\choose3}$$ These ...


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