Binomial theorem for positive integral Index

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Question

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

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Solution

We have,
${(1 + 2 x + x^{2})}^{20} = {\{(1 + x)^{2}\}}^{20} = (1 + x)^{40}$
Clearly, $(1 + x)^{40}$ contains 41 terms.
Hence, ${(1 + 2 x + x^{2})}^{20}$ contains 41 terms.

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