Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 201912 AprilMorning Shift
Exam
JEE Main
Year
2019
Date
12 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2019 · 12 April · Morning Shift · Mathematics · Binomial Theorem

The coefficient of x ^ 18 in the product (1 + x) (1 - x) ^ 10 (1 +...

The coefficient of x18{x}^{18} in the product \left(1 + x\right){\left(1 - x\right)}^{10}{\left(1 + x + {x}^{2}\right)}^{9} is

Options

  1. A8484
  2. B−84- 84
  3. C−126- 126
  4. D126126

Correct answer

A. 8484

Step-by-step solution

(1+x)(1−x)10(1+x+x2)9\left(1 + x\right){\left(1 - x\right)}^{10}{\left(1 + x + {x}^{2}\right)}^{9}
=(1+x)(1−x)(1−x)9(1+x+x2)9= \left(1 + x\right)\left(1 - x\right){\left(1 - x\right)}^{9}{\left(1 + x + {x}^{2}\right)}^{9}
=(1−x2)[(1−x)(1+x+x2)]9= \left(1 - {x}^{2}\right){\left[\left(1 - x\right)\left(1 + x + {x}^{2}\right)\right]}^{9}
=(1−x2)(1−x3)9= \left(1 - {x}^{2}\right){\left(1 - {x}^{3}\right)}^{9}
So, Coefficient of x18in(1+x)(1−x)10(1+x+x2)9{x}^{18} in \left(1 + x\right){\left(1 - x\right)}^{10}{\left(1 + x + {x}^{2}\right)}^{9}
= Coefficient of x18in(1−x2)(1−x3)9{x}^{18} in \left(1 - {x}^{2}\right){\left(1 - {x}^{3}\right)}^{9}
=Coefficient of x18in(1−x2)[C09+C19(−x3)+C2(−x3)2+…9]{x}^{18} in \left(1 - {x}^{2}\right)\left[{C}_{0} 9 + {C}_{1} 9 \left(- {x}^{3}\right) + {C}_{2}{\left(- {x}^{3}\right)}^{2} + \ldots 9\right]
=C69(−1)6=C39=9.8.71.2.3=84= {C}_{6} 9 {\left(- 1\right)}^{6} = {C}_{3} 9 = \dfrac{9.8 . 7}{1.2 . 3} = 84
Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α… JEE Main 2026If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to… JEE Main 2026If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and… JEE Main 2026The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is : JEE Main 2026If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n… JEE Main 2026In the expansion of (9x- 1 3 x )^ 18 , x>0 , if the term independent of x is (221)k… JEE Main 2026Let the smallest value of k ∈ ℕ , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 +… JEE Main 2026If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: JEE Main 2026The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r)… JEE Main 2026Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible… JEE Main 2026The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^… JEE Main 2026Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N… JEE Main 2026The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is : JEE Main 2026The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in… JEE Main 2026Let C_ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n ∈ N, 0… JEE Main 2026The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to JEE Main 2026 Full Binomial Theorem list Binomial Theorem JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs