Home
Class 12
MATHS
Prove that ^n C0^(2n)Cn-^n C1^(2n-2)Cn+^...

Prove that `^n C_0^(2n)C_n-^n C_1^(2n-2)C_n+^n C_2^(2n-4)C_n-=2^ndot`

Text Solution

Verified by Experts

`LHS = C_(0) * ""^(2n)C_(n) - C_(1) *""^(2n-2)C_(n) + C_(2) *""^(2n-4)C_(n)-…`
= Coefficient of ` x^(n) ` in
`[C_(0) (1 +_ x)^(2n) - C_(1) (1 + x)^(2n-2) + C_(2) (1 + x)^(2n-4) -...]`
= Coefficient of ` x^(n) " in " [[(1 +x^(2))^(n)-1^(n)]`
= Coefficient of ` x^(n) " in " (2x + x^(2))^(n)`
= Constant term in ` (2 + x)^(n) = 2^(n)= RHS `
Promotional Banner

Topper's Solved these Questions

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise JEE Type Solved Example : (Matching Type Questions )|2 Videos

  • BIONOMIAL THEOREM

    ARIHANT MATHS|Exercise Exercise For Session 1|8 Videos

  • AREA OF BOUNDED REGIONS

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|23 Videos

  • CIRCLE

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|16 Videos

Similar Questions

Explore conceptually related problems

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-2)C_(n)+^(n)C_(2)^(2n-4)C_(n)-...=2^(n)

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-1)C_(n)+^(n)C_(2)xx^(2n-2)C_(n)++(-1)^(n)sim nC_(n)^(n)C_(n)=1

Prove that ^nC_(0)^(n)C_(0)-^(n+1)C_(1)^(n)C_(1)+^(n+2)C_(2)^(n)C_(2)-...=(-1)^(n)

Prove that : ""^(n)C_(0).""^(2n)C_(n)-""^(n)C_(1).""^(2n-2)Cn_(n)+""^(n)C_(2).""^(2n-4)Cn_(n)+......=2^n

Prove that sum_(r=0)^(2n)(.^(2n)C_(r))^(2)=n^(4n)C_(2n)

Prove that (^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

Prove that C_(0)^(2)+C_(1)^(2)+...C_(n)^(2)=(2n!)/(n!n!)

Prove that (""^(2n)C_(0))^(2)-(""^(2n)C_(1))^(2)+(""^(2n)C_(2))^(2)-…+(""^(2n)C_(2n))^(2)=(-1)^(n)*""^(2n)C_(n) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) + C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

ARIHANT MATHS-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that ^n C0^(2n)Cn-^n C1^(2n-2)Cn+^n C2^(2n-4)Cn-=2^ndot

    Text Solution

    |

  2. The value of (30 0)(30 10)-(30 1)(30 11)+(30 2)(30 12)++(30 20)(30 30)...

    Text Solution

    |

  3. If the coefficient of pth, (p+1)thand (p+2)th terms in the expansion ...

    Text Solution

    |

  4. If the coefficient of x^7 in (ax^2+1/(bx))^11 is equal to the coeffici...

    Text Solution

    |

  5. For natural numbers m ,n ,if(1-y)^m(1+y)^n=1+a1y+a2y^2+ ,a n da1=a2=10...

    Text Solution

    |

  6. In the binomial expansion of (a - b)^(n) , n ge 5 , the sum of the ...

    Text Solution

    |

  7. The sum of the series C 20()0-C 20()1+C 20()2-C 20()3+...-...+C 20()(1...

    Text Solution

    |

  8. Statement-1: sum(r =0)^(n) (r +1)""^(n)C(r) = (n +2) 2^(n-1) Stat...

    Text Solution

    |

  9. The remainder left out when 8^(2n)""(62)^(2n+1) is divided by 9 is (1)...

    Text Solution

    |

  10. For r= 0, 1,.....,10, let Ar,Br, and Crdenote, respectively, the coe...

    Text Solution

    |

  11. So, statement-1 is also true. Stetement-2 is a correct expanation fo...

    Text Solution

    |

  12. The coefficient of x^(7) in the expansion of (1-x-x^(2) + x^(3))^(6) i...

    Text Solution

    |

  13. If n is a positive integer, then (sqrt(3)+1)^(2n)-(sqrt(3)-1)^(2n) is ...

    Text Solution

    |

  14. In the expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x-x^(1/2)))^10 ...

    Text Solution

    |

  15. The coefficients of three consecutive terms of (1+x)^(n+5) are in the ...

    Text Solution

    |

  16. If the coefficients of x^(3) and x^(4) in the expansion of (1 + ax + b...

    Text Solution

    |

  17. Coefficient of x^(11) in the expansion of (1+x^2)(1+x^3)^7(1+x^4)^(12)...

    Text Solution

    |

  18. The sum of coefficients in integral powers of x in the binominal expan...

    Text Solution

    |

  19. The coefficient of x^9 in the expansion of (1+x)(16 x^2)(1+x^3)(1+x^(1...

    Text Solution

    |

  20. If the number of terms in the expansion of (1-2/x+4/(x^2))^n , x!=0, i...

    Text Solution

    |

  21. Let m be the smallest positive integer such that the coefficient of x^...

    Text Solution

    |