The absolute difference of the coefficients of x ^ 10 and x ^ 7 in...

The absolute difference of the coefficients of

{x}^{10} and

{x}^{7} in the expansion of ${\left(2{x}^{2} + \dfrac{1}{2x}\right)}^{11} is equal to

Options

A ${13}^{3} - 13$
B ${11}^{3} - 11$
C ${10}^{3} - 10$
D ${12}^{3} - 12$

Correct answer

D. ${12}^{3} - 12$

Step-by-step solution

The given expansion is ${\left(2{x}^{2} + \dfrac{1}{2x}\right)}^{11}$ .
The general term in the binomial expansion of ${\left(x + a\right)}^{n} is given by$ {T}_{r + 1} = Cr n {x}^{n - r}{a}^{r} $.$ ⇒ {T}_{r + 1} = Cr 11 {\left(2{x}^{2}\right)}^{11 - r}{\left(\dfrac{1}{2x}\right)}^{r} $$ = Cr 11 \left({2}^{11 - r - r}\right){x}^{22 - 2r - r} $Now for coefficient of$ {x}^{7}$ $⇒ 22 - 3r = 7$
$⇒ r = 5$
Coefficient of ${x}^{7} is$ C5 11 \left({2}^{11 - 5 \times 2}\right) $Now for coefficient of$ {x}^{10} ⇒ 22 - 3r = 10 $$ ⇒ r = 4 $Coefficient of$ {x}^{10} is $C4 11 \left({2}^{11 - 2 \times 4}\right)$
We need the absolute difference, we get it by \left|C5 11 \left({2}^{11 - 5 \times 2}\right) - C4 11 \left({2}^{11 - 4 \times 2}\right)\right| $.$ = \left|924 - 2640\right| = 1716 $$ = {12}^{3} - 12 $Hence the absolute difference is$ = {12}^{3} - 12$.

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