Question

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. $30$

Answer 1

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.