The sum of the series 2 . ^ 20 C _ 0 + 5 . ^ 20 C _ 1 + 8 . ^ 20 C...

The sum of the series

{2 .}^{20}{C}_{0} + {5 .}^{20}{C}_{1} + {8 .}^{20}{C}_{2} + {11 .}^{20}{C}_{3} + . . . . . . . + {62 .}^{20}{C}_{20}

is equal to

Options

A ${2}^{26}$
B ${2}^{25}$
C ${2}^{24}$
D ${2}^{23}$

Correct answer

B. ${2}^{25}$

Step-by-step solution

Let,

$S = 2{.}^{20}{C}_{0} + 5{.}^{20}{C}_{1} + 8{.}^{20}{C}_{3} + . . . . . . . + 62{.}^{20}{C}_{20} . . .$ \left(i\right)
$S = 62{.}^{20}{C}_{0} + 59{.}^{20}{C}_{1} + 56{.}^{20}{C}_{3} + . . . . . . . + 2{.}^{20}{C}_{20} . . .$ \left(ii\right) $(Writing in reverse order$ n{C}_{r}{=}^{n}{C}_{n - r} $)<br />Adding \left(i\right) and \left(ii\right)$
$2S = 64{(}^{20}{C}_{0}{+}^{20}{C}_{1}{+}^{20}{C}_{2} + . . . . . {+}^{20}{C}_{20} )$
$⇒ S = {32.2}^{20} = {2}^{25}$

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