Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202126 AugustMorning Shift
Exam
JEE Main
Year
2021
Date
26 August
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2021 · 26 August · Morning Shift · Mathematics · Binomial Theorem

If Cr 20 is the co-efficient of x ^ r in the expansion of ( 1 + x )...

If Cr20Cr 20 is the co-efficient of xrintheexpansionof{x}^{r} in the expansion of( 1 + x{)}^{20},thenthevalueof, then the value of\mathop{∑}_{r = 0}^{20}{r}^{2}(Cr20)\left(Cr 20\right) is equal to:

Options

  1. A420×218420 \times {2}^{18}
  2. B380×218380 \times {2}^{18}
  3. C380×219380 \times 219
  4. D420×219420 \times {2}^{19}

Correct answer

A. 420×218420 \times {2}^{18}

Step-by-step solution

∑r=020r2(Cr20)=∑r(r⋅Cr20)=∑r=120r⋅20.Cr−119\mathop{∑}_{r = 0}^{20}{r}^{2}\left(Cr 20\right) = ∑ r\left(r \cdot Cr 20\right) = \mathop{∑}_{r = 1}^{20}r \cdot 20 . Cr - 1 19

⇒20∑r=120(r−1+1)⋅19Cr−1=20(∑r=120(r−1)⋅19Cr−1+∑r=120Cr−119)⇒ 20\mathop{∑}_{r = 1}^{20} ( r - 1 + 1) {\cdot}^{19}{C}_{r - 1} = 20\left(\mathop{∑}_{r = 1}^{20} ( r - 1 ) {\cdot}^{19}{C}_{r - 1} + \mathop{∑}_{r = 1}^{20} Cr - 1 19\right)

⇒20(19∑r=220Cr−218+∑r=120Cr−119)⇒ 20\left(19\mathop{∑}_{r = 2}^{20} Cr - 2 18 + \mathop{∑}_{r = 1}^{20} Cr - 1 19\right)

⇒(380)218+20⋅219⇒220⋅(95+10)=(105)220=420×218⇒ ( 380 ) {2}^{18} + 20 \cdot {2}^{19} ⇒ {2}^{20} \cdot ( 95 + 10 ) = ( 105 ) {2}^{20} = 420 \times {2}^{18}

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