Question

The number of terms in the expression of ${(1 + x)^{101}}{(1 + {x^2} - x)^{100}}$ in the power of x is:-
A. 302
B. 301
C. 202
D. 101

Answer 1

Hint: Here in this question algebraic identities, binomial expansion and some surds and indices concept will get used which are mentioned below:-
*$({a^3} + {b^3}) = (a + b)({a^2} + {b^2} - ab)$
*$({a^x} \times {a^y}) = {(a)^{x + y}}$
*$({a^x} \times {b^x}) = {(ab)^x}$

Complete step-by-step answer:
Now first of all we will try to simplify the given equation.$ \Rightarrow {(1 + x)^{101}}{(1 + {x^2} - x)^{100}}$
We will split ${(1 + x)^{101}}$into two parts such that one part should have degree 100\[ \Rightarrow (1 + x){(1 + x)^{100}}{(1 + {x^2} - x)^{100}}\] (We have applied identity$({a^x} \times {a^y}) = {(a)^{x + y}}$)\[ \Rightarrow (1 + x){[(1 + x)(1 + {x^2} - x)]^{100}}\] (We have applied identity$({a^x} \times {b^x}) = {(ab)^x}$)
Now we know$({a^3} + {b^3}) = (a + b)({a^2} + {b^2} - ab)$ so to simplify equation more we will apply this identity as mentioned below:-
$ \Rightarrow ({1^3} + {x^3}) = (1 + x)({1^2} + {x^2} - x)$
Now we will put this simplified term in the equation.
\[ \Rightarrow (1 + x){[(1 + {x^3})]^{100}}\] .................Equation (1)
We know that binomial expansion is given by ${(1 + x)^n} = \sum\limits_{r = 0}^n {^n{C_r}.{X^r} = [{C_0} + {C_1}X + {C_2}{X^2} + ...{C_n}{X^n}]} $Now we will put value of n=100 in the expansion
$ \Rightarrow {(1 + {x^3})^{100}} = [{C_0} + {C_1}{x^3} + {C_2}{x^6} + ...{C_{100}}{x^{300}}]$
Now we will put this expansion value in equation 1
\[ \Rightarrow (1 + x)[{C_0} + {C_1}{x^3} + {C_2}{x^6} + ...{C_{100}}{x^{300}}]\]
Now multiplying whole expansion with 1 and then with x\[ \Rightarrow [{C_0} + {C_1}{x^3} + {C_2}{x^6} + ...{C_{100}}{x^{300}}][x{C_0} + {C_1}{x^4} + {C_2}{x^7} + ...{C_{100}}{x^{301}}]\]
As all the terms are different so they cannot be clubbed together and hence cannot be simplified further. Therefore terms in first expansion are 101 and other terms will also contain 101 terms, so the total terms in power of x in expansion will be 202.

So, the correct answer is “Option C”.

Note: Students may likely to make mistake while solving the last step of this question because no further simplification can be done on two different terms for example if we have equation like $3{x^2} + 4{x^2} + 2x$ then if we count the terms it would be two because same power can be added with each other but not with some other power. So the final equation will be $7{x^2} + 2x$ which contains two terms.