Find the number of terms in the expansion of ${(1 + 2 x + x^{2})}^{20}$

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Solution

$\begin{matrix} {(1 + 2 x + x^{2})}^{20} = {(1 + x)}^{2} s o, {({(1 + x)}^{2})}^{20} = {(1 + x)}^{40} s o f r o m b i m o n i a l e x p a n s i o n, w e h a v e^{40} C_{0} x^{40} +^{40} C_{1} x^{39} . . . . . . . . . .^{40} C_{40} x^{0} s o t o t a l t e r m i n t h e e x p a n s i o n w i l l b e 41 (40 t e r m w i l l b e t h e r e f r o m^{40} C_{1} t o^{40} C_{40} + 1 t e r m^{40} C_{0} s o t o t a l t e r m w i l l b e 41) \end{matrix}$

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