+26400 Evaluate without a calculator:
\([~12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3} + \cdots + 2\binom{13}{3} +\binom{14}{3} ~] \)
Identity:
\(\text{For }~ n,r \in N^+, r \le n,\\ \dbinom{n+1}{r+1} = \sum \limits_{j=r}^{n} \dbinom{j}{r}\)
see: http://www.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Pascal's_triangle.html
\(\small{ \begin{array}{lcl} \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3}+ \binom{8}{3}+ \binom{9}{3}+ \binom{10}{3}+ \binom{11}{3}+ \binom{12}{3} + \binom{13}{3} +\binom{14}{3} &=& \binom{15}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3}+ \binom{8}{3}+ \binom{9}{3}+ \binom{10}{3}+ \binom{11}{3}+ \binom{12}{3} + \binom{13}{3} &=& \binom{14}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3}+ \binom{8}{3}+ \binom{9}{3}+ \binom{10}{3}+ \binom{11}{3}+ \binom{12}{3} &=& \binom{13}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3}+ \binom{8}{3}+ \binom{9}{3}+ \binom{10}{3}+ \binom{11}{3} &=& \binom{12}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3}+ \binom{8}{3}+ \binom{9}{3}+ \binom{10}{3} &=& \binom{11}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3}+ \binom{8}{3}+ \binom{9}{3} &=& \binom{10}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3}+ \binom{8}{3} &=& \binom{9}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3}+ \binom{7}{3} &=& \binom{8}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3}+ \binom{6}{3} &=& \binom{7}{4} \\ \binom{3}{3} + \binom{4}{3} + \binom{5}{3} &=& \binom{6}{4} \\ \binom{3}{3} + \binom{4}{3} &=& \binom{5}{4} \\ \binom{3}{3} &=& \binom{4}{4} \end{array} } \)
\(\small{ \begin{array}{lcl} \binom{4}{4} +\binom{5}{4}+\binom{6}{4}+ \binom{7}{4}+ \binom{8}{4}+\binom{9}{4}+\binom{10}{4}+\binom{11}{4}+\binom{12}{4}+\binom{13}{4}+\binom{14}{4}+\binom{15}{4} \\ = \binom{16}{5}\\ 12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3}+9\binom{6}{3}+8\binom{7}{3}+7\binom{8}{3}+6\binom{9}{3}+5\binom{10}{3}+4\binom{11}{3}+3\binom{12}{3} + 2\binom{13}{3} +\binom{14}{3} \\ = \binom{16}{5} \\ 12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3}+9\binom{6}{3}+8\binom{7}{3}+7\binom{8}{3}+6\binom{9}{3}+5\binom{10}{3}+4\binom{11}{3}+3\binom{12}{3} + 2\binom{13}{3} +\binom{14}{3} \\ = \frac{16}{5}\cdot \frac{15}{4}\cdot \frac{14}{3}\cdot \frac{13}{2}\cdot \frac{12}{1} \\\\ \mathbf{ 12\binom{3}{3} + 11\binom{4}{3} + 10\binom{5}{3} +9\binom{6}{3}+8\binom{7}{3}+7\binom{8}{3}+6\binom{9}{3} +5\binom{10}{3}+4\binom{11}{3}+3\binom{12}{3} + 2\binom{13}{3} +\binom{14}{3} } \\ \mathbf{=} \mathbf{4368 } \end{array} } \)
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