Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202522 Jan 2025Evening ShiftMathematicsBinomial TheoremActual

If _ r=1 ³⁰ r^2 ( ³⁰ C_r )^2 30 r-1 = 2²⁹ , then is equal to ______

Correct answer

0

Step-by-step solution

aligned & _ r =1 ³⁰ r ^2 ( ³⁰ C _ r )^2 ³⁰ C _ r -1 & = _ r =1 ³⁰ r ^2 ( 31- r r ) 30! r !(30- r )! & ( ³⁰ C _ r ³⁰ C _ r -1 = 30- r +1 r = 31- r r ) aligned aligned & = _ r =1 ³⁰ (31- r ) 30! ( r -1)!(30- r )! & =30 _ r =1 ³⁰ (31- r ) 29! ( r -1)!(30- r )! & =30 _ r =1 ³⁰(30- r +1)²⁹ C _ 30- r & =30 ( _ r =1 ³⁰(31- r )²⁹ C _ 30- r + _ r =1 ³⁰ ²⁹ C _ 30- r ) & =30 (29 2²⁸+2²⁹ )=30(29+2) 2²⁸ & =15 31 2²⁹ & =465 (2²⁹ ) & =465 aligned

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