[tex] C_n^k = C_{n-1}^k+C_{n-1}^{k-1} \rightarrow formula \\ \\ C_{20}^{10} = \\ \\ = C_{19}^{10}+ C_{19}^{9} \\ \\= C_{18}^{10} + C_{18}^{9}+ C_{19}^{9} \\ \\ = C_{17}^{10} + C_{17}^{9} + C_{18}^{9}+ C_{19}^{9} \\ .\\.\\.\\= C_{10}^{10} + C_{10}^9 + C_{11}^9 +...+C_{19}^9 \\ \\ = C_{9}^9+C_{10}^9 + C_{11}^9 +...+C_{19}^9 [/tex]
