Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202226 Jul 2022Evening ShiftMathematicsBinomial TheoremActual

∑ i , j = 0 i ≠ j n C i n C j n is equal to

Options

  1. A2 2 n - C n 2 n
  2. B2 2 n - 1   - C n - 1 2 n - 1
  3. C2 2 n - 1 2 C n 2 n
  4. D2 n - 1 + C n 2 n - 1

Correct answer

A. 2 2 n - C n 2 n

Step-by-step solution

∑ i , j = 0 i ≠ j n C i n C j n = C 0 n C 1 n + C 2 n + . . . . C n n + C 1 n C 0 n + C 2 n + . . . . C n n + . . . . C n n C 0 n + C 1 n + . . . . C n - 1 n = 2 C 0 n C 1 n + C 1 n C 2 n + . . . . C n - 1 n C n n = C 0 n + C 1 n . . . . C n n 2 - C 0 n 2 + C 1 n 2 . . . . . C n n 2 = 2 n 2 - C n 2 n = 2 2 n - C n 2 n

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 + + 2¹³ 13 12 12 ) = 3¹³ - , then is equal to: 2026If (1 - x^3)¹⁰ = _ r=0 ¹⁰ a_r x^r (1-x)^ 30-2r , then 9a₉ a₁₀ is equal to __________. 2026If the coefficients of the middle terms in the binomial expansions of (1 + x)²⁶ and (1 - x)²⁸ , 0 , are equal, then the value of is: 2026The coefficient of x^2 in the expansion of (2x^2 + 1 x )¹⁰ , x 0 , is : 2026If the sum of the coefficients of x^7 and x¹⁴ in the expansion of ( 1 x^3 - x^4 )^n , x 0 , is zero, then the value of n is __________. 2026In the expansion of (9x- 1 3 x )¹⁸ , x>0 , if the term independent of x is (221)k , then k is equal to: 2026Let the smallest value of k N , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 + (1+x)^5 + + (1+x)⁹⁹ + (1+kx)¹⁰⁰ , x 0 , is (43n + 101 4 ) (¹⁰⁰C₃ ) for some n N , be p . The 2026If for 3 r 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: 2026 Full Binomial Theorem list All JEE Main PYQs