Home
Class 12
MATHS
Evaluate sum(i=0)^(n-1) sum(j= 1 + i)^(n...

Evaluate `sum_(i=0)^(n-1) sum_(j= 1 + i)^(n+1)""^(n)C_(i) ""^(n+1)C_(j)` .

Text Solution

Verified by Experts

Let ` P = sum_(i=0)^(n-1) sum_(j= 1 + i)^(n+1)""^(n)C_(i) ""^(n+1)C_(j)` .
`sum_(j=1)^(n-1)""^(n)C_(0)""^(n+1)C_(j) + sum_(j= 2)^(n+1)""^(n)C_(i) ""^(n+1)C_(j)+ sum_(j= 3)^(n+1) ""^(n)C_(2)""^(n+1)C_(j) + ...+ sum_(j = n)^(n+1) ""^(n)C_(n-1) ""^(n+1)C_(j)`
`""^(n)C_(0) sum_(j=1)^(n-1)""^(n)C_(j)+""^(n)C_(1) + sum_(j= 2)^(n+1)""^(n+1)C_(j) ""^(n+1)C_(2)+ sum_(j= 3)^(n+1) ""^(n+1)C_(j) + ...+ ""^(n)C_(n-1)sum_(j = n)^(n+1) ""^(n+1)C_(j)`
`""^(n)C_(0)(""^(n +1)C_(j) + ""^(n+1)C_(2) + ""^(n+1)C_(3) + ...+ ""^(n+1)C_(n+1))`
` + ""^(n)C_(1) (""^(n+1)C_(2)+""^(n+1)C_(3) + ""^(n+1)C_(4)+ ...+ ""^(n+1)C_(n+1))`
` + ""^(n)C_(2) (""^(n+1)C_(3)+""^(n+1)C_(4) + ""^(n+1)C_(5)+ ...+ ""^(n+1)C_(n+1))`
` + ...+ ""^(n)C_(n-1)(""^(n +1)C_(n) + ""^(n+1)C_(n+1))`
`= ""^(n+1)C_(1)*""^(n)C_(0) + ""^(n+1)C_(2) (""^(n)C_(0) + ""^(n)C_(1)) + ""^(n+1)C_(3) (""^(n)C_(0) + ""^(n)C_(1) + ""^(n)C_(2))`
` + ...+ ""^(n+1)C_(n+1) (""^(n)C_(0) + ""^(n)C_(1) + ""^(n)C_(2)+ ...+ ""^(n)C_(n-1))`
`(""^(n)C_(0) + ""^(n)C_(1)) *""^(n)C_(0)+ (""^(n)C_(1) + ""^(n)C_()) ( ""^(n)C_(2)+ ""^(n)C_(1))+ (""^(n)C_(2) + ""^(n)C_(3)) (""^(n)C_(0) + ""^(n)C_(1) + ""^(n)C_(2))`
`+ ...+ (""^(n)C_(n) + ""^(n)C_(n-1))(""^(n)C_(0) + ""^(n)C_(1)+ ""^(n)C_(2) + ...+ ""^(n)C_(n-1)) + n`
`(""^(n)C_(0))^(2) + (""^(n)C_(1))^(2) + (""^(n)C_(2))^(2) + ...+ (""^(n)C_(n-1))^(2)`
` + {""^(n)C_(0) *""^(n)C_(1) + ""^(n)C_(0) * ""^(n)C_(2) + ""^(n)C_(0) *""^(n)C_(3)`
`+...+""^(n)C_(0) * ""^(n)C_(n-1) + ...+ ""^(n)C_(n-2) + ""^(n)C_(n-1)} + 2^(n) - 1 + n`
` (""^(n)C_(0) + ""^(n)C_(1) + ""^(n)C_(2) + ...+ ""^(n)C_(n-1))^(2) + 2^(n) - 1 + n`
`= (2^(n) -1)^(2) + 2^(n) -1 + n = 2 ^(2n) - 2^(n) + n`
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

Evaluate sum_(i=0)^(n)sum_(j=0)^(n) ""^(n)C_(j) *""^(j)C_(i), ile j .

S=sum_(i=1)^(n)sum_(j=1)^(i)sum_(k=1)^(j)1

sum_(i=0)^(n)sum_(j=0)^(m)*^(n)C_(i)*C_(j) is equal to

sum_(j=1)^(n)sum_(i=1)^(n)i=

The value of sum_(i=1)^(n) sum_(j=1)^(i) sum_(k=1)^(j) 1 is

sum_ (i = 0) ^ (n) (- 1) ^ (n) (n + 1) * (n) C_ (n) 3 ^ (n)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) +…+ C_(n) x^(n) , find the values of the following . sum_(i=0)^(n) sum_(j=0)^(n) (i+j) C_(i) C_(j)

Find the sum sum_(0<=i

sum_(i=1)^(n) sum_(i=1)^(n) i is equal to

If sum_(i=1)^(n-1) ((""^(n)C_(i-1))/(""^(n)C_(i)+""^(n)C_(i-1)))^(3) = (36)/(13) , , then n is equal to

ARIHANT MATHS-BIONOMIAL THEOREM-Exercise (Questions Asked In Previous 13 Years Exam)
  1. Evaluate sum(i=0)^(n-1) sum(j= 1 + i)^(n+1)""^(n)C(i) ""^(n+1)C(j) .

    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

    |