The number of terms in the expansion of ${\left(1+2x+x^2\right)}^{20}$ when expanded in descending powers of $x$, is 

We have, 

${\left(1+2x+x^2\right)}^{20}={\left\{(1+x{)}^2\right\}}^{20}=(1+x{)}^{40}$

Clearly, $(1+x{)}^{40}$ contains 41 terms.

Hence, ${\left(1+2x+x^2\right)}^{20}$ contains 41 terms.