Combinatoric Lovers

$S_{m,n} = \dbinom{n}{m}+2 \dbinom{n-1}{m}+3 \dbinom{n-2}{m}+\cdots+ (n-m+1)\dbinom{m}{m} \\ S_{m,n}=\left[ \dbinom{n}{m}+ \dbinom{n-1}{m}+\dbinom{n-2}{m}+\cdots+\dbinom{m}{m} \right] + \left[ \dbinom{n-1}{m}+\dbinom{n-1}{m}+\dbinom{n-2}{m}+\cdots+\dbinom{m}{m} \right] \\ \quad \quad \quad +\left[ \dbinom{n-2}{m}+ \dbinom{n-1}{m}+ \dbinom{n-2}{m}+\cdots+\dbinom{m}{m} \right] +\cdots\cdots+\left[ \dbinom{m+1}{m}+\dbinom{m}{m} \right] +\dbinom{m}{m} \\ S_{m,n} = \dbinom{n+1}{m+1}+ \dbinom{n}{m+1}+\dbinom{n-1}{m+1}+\cdots+\dbinom{m+1}{m+1} \quad \small\color{#3D99F6}{\dbinom{m+1}{m+1}=\dbinom{m}{m}}$ $\large \boxed{S_{m,n} = \dbinom{n+2}{m+2} }$

$\text{For } n=25,m=5 \\ S_{5,25} = \dbinom{27}{7} = 888030$

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$\text{For visualization take an example of } n=3 \text{ and } m=0 \\ S_{0,3} =\dbinom{3}{0}+2\dbinom{2}{0}+3\dbinom{1}{0}+4\dbinom{0}{0} = 1+2+3+4 = \dbinom{4}{1}+\dbinom{3}{1}+\dbinom{2}{1}+\dbinom{1}{1} = \dbinom{5}{2} \\ \text{take an example of } n=5 \text{ and } m=3 \\ S_{3,5} =\dbinom{5}{3}+2\dbinom{4}{3}+3\dbinom{3}{3} = \dbinom{6}{4}+\dbinom{5}{4}+\dbinom{4}{4} = \dbinom{7}{5} = 35$

#Hockey Stick Identity