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\begin{align*} \binom{5}{2} &= \binom{4}{2} + \binom{4}{1} \\ &= (\binom{3}{2} + \binom{3}{1}) + (\binom{3}{1} + \binom{3}{0})\\ &= \binom{3}{2} + 2\binom{3}{1}) + 1 \\ &= (\binom{2}{2} + \binom{2}{1}) + 2(\binom{2}{1} + \binom{2}{0}) + 1\\ &= (1 + \binom{2}{1}) + 2\binom{2}{1} + 1) + 1 \\ &= 3\binom{2}{1} + 4 \\ &= 3(\binom{1}{1} + \binom{1}{0}) + 4 \\ &= 3(1+1) + 4 = 10 \end{align*}