+26382 There exist positive integers n and k such that
\(32 \dbinom{6}{6} + 31 \dbinom{7}{6} + 30 \dbinom{8}{6} + \dots + 3 \dbinom{35}{6} + 2 \dbinom{36}{6} + \dbinom{37}{6} = \dbinom{n}{k}\)
Enter the ordered pair \((n, k)\)
Using the hockey stick identity: \(\sum \limits_{i=r}^{n} \dbinom{i}{r} = \dbinom{n+1}{r+1} \qquad \text{for } n,r\in \mathbb{N},\quad n \geq r\)
see: https://en.wikipedia.org/wiki/Hockey-stick_identity
\(\begin{array}{|rcll|} \hline &&\mathbf{ 32 \dbinom{6}{6} + 31 \dbinom{7}{6} + 30 \dbinom{8}{6} + \dots + 3 \dbinom{35}{6} + 2 \dbinom{36}{6} + \dbinom{37}{6} } \\\\ &=& \sum \limits_{i=6}^{37} \dbinom{i}{r} +\sum \limits_{i=6}^{36} \dbinom{i}{r} +\sum \limits_{i=6}^{35} \dbinom{i}{r} +\sum \limits_{i=6}^{34} \dbinom{i}{r} +\sum \limits_{i=6}^{33} \dbinom{i}{r} +\sum \limits_{i=6}^{32} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{31} \dbinom{i}{r} +\sum \limits_{i=6}^{30} \dbinom{i}{r} +\sum \limits_{i=6}^{29} \dbinom{i}{r} +\sum \limits_{i=6}^{28} \dbinom{i}{r} +\sum \limits_{i=6}^{27} \dbinom{i}{r} +\sum \limits_{i=6}^{26} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{25} \dbinom{i}{r} +\sum \limits_{i=6}^{24} \dbinom{i}{r} +\sum \limits_{i=6}^{23} \dbinom{i}{r} +\sum \limits_{i=6}^{22} \dbinom{i}{r} +\sum \limits_{i=6}^{21} \dbinom{i}{r} +\sum \limits_{i=6}^{20} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{19} \dbinom{i}{r} +\sum \limits_{i=6}^{18} \dbinom{i}{r} +\sum \limits_{i=6}^{17} \dbinom{i}{r} +\sum \limits_{i=6}^{16} \dbinom{i}{r} +\sum \limits_{i=6}^{15} \dbinom{i}{r} +\sum \limits_{i=6}^{14} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{13} \dbinom{i}{r} +\sum \limits_{i=6}^{12} \dbinom{i}{r} +\sum \limits_{i=6}^{11} \dbinom{i}{r} +\sum \limits_{i=6}^{10} \dbinom{i}{r} +\sum \limits_{i=6}^{9} \dbinom{i}{r} +\sum \limits_{i=6}^{8} \dbinom{i}{r} \\ && +\sum \limits_{i=6}^{7} \dbinom{i}{r} +\sum \limits_{i=6}^{6} \dbinom{i}{r} \\ \\ &=& \dbinom{38}{7} +\dbinom{37}{7} +\dbinom{36}{7} +\dbinom{35}{7} +\dbinom{34}{7} +\dbinom{33}{7} \\ && \dbinom{32}{7} +\dbinom{31}{7} +\dbinom{30}{7} +\dbinom{29}{7} +\dbinom{28}{7} +\dbinom{27}{7} \\ && \dbinom{26}{7} +\dbinom{25}{7} +\dbinom{24}{7} +\dbinom{23}{7} +\dbinom{22}{7} +\dbinom{21}{7} \\ && \dbinom{20}{7} +\dbinom{19}{7} +\dbinom{18}{7} +\dbinom{17}{7} +\dbinom{16}{7} +\dbinom{15}{7} \\ && \dbinom{14}{7} +\dbinom{13}{7} +\dbinom{12}{7} +\dbinom{11}{7} +\dbinom{10}{7} +\dbinom{9}{7} \\ && +\dbinom{8}{7} +\dbinom{7}{7} \\\\ &=& \sum \limits_{i=7}^{38} \dbinom{i}{r} \\\\ &=& \mathbf{\dbinom{39}{8}} \\ \hline \end{array}\)
