Home
Class 11
MATHS
Prove that ^nC0 "^(2n)Cn-^nC1 ^(2n-2)Cn ...

Prove that `^nC_0 "^(2n)C_n-^nC_1 ^(2n-2)C_n +^nC_2 ^(2n-4)C_n =2^n`

Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS AND QUADRATIC EQUATIONS

    CENGAGE PUBLICATION|Exercise All Questions|877 Videos

Similar Questions

Explore conceptually related problems

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) *""^(2n)C_(n) - C_(1) *""^(2n-2)C_(n) + C_(2) *""^(2n-4) C_(n) -…= 2^(n)

If (18x^2+12x+4)^n = a_0 +a_(1x)+ a_(2x)^2 +......+ a_(2n)x^(2n) , prove that a_r= 2^n3^r ( "^(2n)C_r + "^(n)C_1 "^(2n-2)C_r + "^(n)C_2 "^(2n-4)C_r + ....

Prove that .^n C_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 .^n C_0-^(n+1)C_1 . ^n C_1+^(n+2)C_2 . ^n C_2-=(-1)^ndot

Prove that (.^(2n)C_0)^2-(.^(2n)C_1)^2+(.^(2n)C_2)^2-..+(.^(2n)C_(2n))^2 = (-1)^n.^(2n)C_n .

show that ^nC_r+ ^(n-1)C_(r-1)+ ^(n-1)C_(r-2)= ^(n+1)C_r

If ninNN , then by principle of mathematical induction prove that, .^(n)C_(0)+.^(n)C_(1)+.^(n)C_(2)+ . . .+.^(n)C_(n)=2^(n)(ninNN)

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

Prove that (2nC_0)^2-(2nC_1)^2+(2nC_2)^2+.....+(2nC_2n)^2=(-1)^n2nC_n

If (1+x)^(n)=^(n)C_(0)+^(n)C_(1)x+^(n)C_(2)x^(2)+…+^(n)C_(n)x^(n) , prove that, nC_(0) +2^(n)C_(1)+3^(n)C_(2)+…+(n+1)^(n)C_(n)=(n+2)*2^(n-1) .

CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. Find the value of x , for which 1/(sqrt(5+4x)) can be expanded as infi...

    Text Solution

    |

  2. Find the fourth term in the expansion of (1-2x)^(3//2)""dot

    Text Solution

    |

  3. Prove that ^nC0 "^(2n)Cn-^nC1 ^(2n-2)Cn +^nC2 ^(2n-4)Cn =2^n

    Text Solution

    |

  4. Prove that ^n C0 .^n C0-^(n+1)C1 . ^n C1+^(n+2)C2 . ^n C2-=(-1)^ndot

    Text Solution

    |

  5. The sum of the coefficients of all the integral powers of x in the exp...

    Text Solution

    |

  6. If the sum of the coefficient in the expansion of (alpha^2x^2-2alphax+...

    Text Solution

    |

  7. Prove that sum(alpha+beta+gamma = 10) (10 !)/(alpha!beta!gamma!)=3^(10...

    Text Solution

    |

  8. If (1+x-2x^2)^(20)=a0+a1x+a2x^2+a3x^3+...+a(40)x^(40), then find the v...

    Text Solution

    |

  9. Find the sum of the series .^15 C0+^(15)C1+^(15)C2+...............+^(1...

    Text Solution

    |

  10. Find the sum sum(k=0)^(10).^(20)Ck.

    Text Solution

    |

  11. Find the sum of all the coefficients in the binomial expansion of (x^2...

    Text Solution

    |

  12. If the sum of coefficient of first half terms in the expansion of (x+y...

    Text Solution

    |

  13. Find the value of underset(p=1)overset(n)sum(underset(m=p)overset(n)su...

    Text Solution

    |

  14. Show that the middle term in the expansion of (x+1)^(2n) is (1.3.5.......

    Text Solution

    |

  15. If the middle term in the expansion of (x^2+1//x)^n is 924 x^6 , then...

    Text Solution

    |

  16. The first three terms in the expansion of (1+a x)^n(n!=0) are 1,6x and...

    Text Solution

    |

  17. If x^4 occurs in the rth term in the expansion of (x^4+1/(x^3))^(15), ...

    Text Solution

    |

  18. Find the coefficient of x^(-10) in the expansion of (a/x+b x)^(12) .

    Text Solution

    |

  19. Find the constant term in the expansion of (x-1/x)^6dot

    Text Solution

    |

  20. If the coefficients of (r-5) th and(2r-1) th terms in the expansion of...

    Text Solution

    |