Theo nhị thức Niu-tơn ta có:\({\left( {1 + i} \right)^{19}} = (C_{19}^0 + C_{19}^2{i^2} + C_{19}^4{i^2} + ... + C_{19}^{16}{i^2} + C_{19}^{18}{i^2}) + (C_{19}^1i + C_{19}^3{i^3} + ... + C_{19}^{19})\)Phần thực ở vế phải là: \(C_{19}^0 - C_{19}^2 + C_{19}^4 - ... + C_{19}^{16} - C_{19}^{18}.\)Mặt khác:\(\eqalign{
& {\left( {1 + i} \right)^{19}} = {\left[ {\sqrt 2 \left( {\cos {\pi \over 4} + i\sin {\pi \over 4}} \right)} \right]^{19}} = {\left( {\sqrt 2 } \right)^{19}}\left( {\cos {{19\pi } \over 4} + i\sin {{19\pi } \over 4}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {\sqrt 2 } \right)^{19}}\left( { - {{\sqrt 2 } \over 2} + i{{\sqrt 2 } \over 2}} \right) = - {2^9} + {2^9}i \cr
& \Rightarrow C_{19}^0 - C_{19}^2 + C_{19}^4 - ... + C_{19}^{16} - C_{19}^{18} =- {2^9} = - 512. \cr} \)
