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How many terms are there in the expansion of \[{{\text{(1 + 2x + }}{{\text{x}}^{\text{2}}}{\text{)}}^{{\text{10}}}}\]
A. \[11\]
B. \[20\]
C. \[21\]
D. \[30\]

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Last updated date: 13th Jun 2024
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Answer
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Hint: Using the formula of \[{{\text{(a + b)}}^{\text{2}}}{\text{ = }}{{\text{a}}^{\text{2}}}{\text{ + }}{{\text{b}}^{\text{2}}}{\text{ + 2ab}}\] in the above question and also applying the binomial expansion in the above question we will get the required form of the equation. Also use the concept that for \[{{\text{(a + b)}}^{\text{n}}}\]total number of terms are \[{\text{n + 1}}\].

Complete step by step answer:

As per the given equation is \[{{\text{(1 + 2x + }}{{\text{x}}^{\text{2}}}{\text{)}}^{{\text{10}}}}\]
As first observing the above given equation and converting it into the simpler form as
\[{\text{1 + 2x + }}{{\text{x}}^{\text{2}}}\]= \[{{\text{x}}^{\text{2}}}{\text{ + }}{{\text{1}}^{\text{2}}}{\text{ + 2(1)(x)}}\]=\[{{\text{(x + 1)}}^2}\]
And hence replacing it in the above equation, we get,
\[{{\text{(1 + 2x + }}{{\text{x}}^{\text{2}}}{\text{)}}^{{\text{10}}}} = {{\text{(x + 1)}}^{20}}\]
And hence using the above concept of for \[{{\text{(a + b)}}^{\text{n}}}\]total number of terms are \[{\text{n + 1}}\].
So, the total number of terms will be \[21\]
Hence, option (c) is our correct answer.

Note:The Binomial theorem states us the way and definite procedure of expanding expressions of the form \[{{\text{(a + b)}}^{\text{n}}}\]. A binomial has two terms. By definition, a binomial is a polynomial with exactly two terms. The bottom number of the binomial coefficient is \[{\text{r - 1}}\], where r is the term number. a is the first term of the binomial and its exponent is n – (\[{\text{r - 1}}\]), where n is the exponent on the binomial and r is the term number. b is the second term of the binomial and its exponent is \[{\text{r - 1}}\], where r is the term number.