Home
Class 12
MATHS
If (1+x)^n=C0+C1x+C2x^2+...+Cn x^n , the...

If `(1+x)^n=C_0+C_1x+C_2x^2+...+C_n x^n ,` then `C_0C_2+C_1C_3+C_2C_4+...+C_(n-2)C_n=` a. `((2n)!)/((n !)^2)` b. `((2n)!)/((n-1)!(n+1)!)` c. `((2n)!)/((n-2)!(n+2)!)` d. none of these

A

`((2n)!)/((n!)^(2))`

B

`((2n)!)/((n-1)!(n+1)^(!))`

C

`((2n)!)/((n-2)!(n+2)!)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
`(1+x)^(n)=C_(0)+.^(n)C_(1)x+.^(n)C_(2)x^(2)+.^(n)C_(3)x^(3)+"....."+C_(n-1)x^(n-1)+C_(n)x^(n) " "(1)`
`(x+1)^(n) = C_(0)x^(n)+C_(1)x^(n-1)+C_(2)x^(n-2)+"...."+C_(n-1)x+C_(n)" "(2)`
Multiplying Eqs. (1) and (2) and equating the coefficient of `x^(n-2)`. we get
`C_(0)C_(2)+C_(1)C_(3)+C_(2)C_(4)+"....."C_(n-2)C_(n)`
`=` coefficient of `x^(n-2)` in `(1+x)^(2n)`
`= .^(2n)C_(n-2)`
`= ((2n)!)/((n-2)!(n+2)!)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer Type|27 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Linked Comphrension|20 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.8|10 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

Similar Questions

Explore conceptually related problems

If (1+x)^n=C_0+C_1x+C_2x^2+…..+C_nx^n .then show that C_1+2C_2+…. nC_n=n.2^(n-1) .

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+.....+C_(n)x^(n) then show : C_(0)C_(1)+C_(1)C_(2)+C_(2)C_(3)+.....+C__(n-1)C_(n)=((2n)!)/((n+1)!(n-1)!)

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+.....+C_(n)x^(n) then show : C_(0).C_(n)+C_(1).C_(n-1)+C_(2).C_(n-2)+....+C_(n).C_(0)=((2n)!)/((n!)^(2))

If (1+x)^n=C_0+C_1x+C_2 x^2+...+C_n x^n , n in N ,then C_0-C_1+C_2-.....+(-1)^(n-1)C_(m-1), is equal to (mltn)

If (1+x)^(n)=C_(0)+C_(1)+x+C_(2)x^(2)+...+C_(n) x^(n) Show that C_(1)^(2)+2*C_(2)^(2)+3*C_(3)^(2)....+n*C_(n)^(2)=((2n-1)!)/([(n-1)!]^(2))

If (1+x)^n=c_0+c_1x+c_2x^2+....+c_nx^n ,then show that c_0/1+c_2/3+c_4/5+...=2^n/(n+1)

The coefficient of 1//x in the expansion of (1+x)^n(1+1//x)^n is (a). (n !)/((n-1)!(n+1)!) (b). ((2n)!)/((n-1)!(n+1)!) (c). ((2n)!)/((2n-1)!(2n+1)!) (d). none of these

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+.....+C_(n)x^(n) then show : C_(1)+2.C_(2)+3.C_(3)+....+n.C_(n)=n.2^(n-1)

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+.....+C_(n)x^(n) then show : (C_(1))/(C_(0))+(2C_(2))/(C_(1))+(3C_(3))/(C_(2))+....+(nC_(n))/(C_(n-1))=(n(n-1))/(2)

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+.....+C_(n)x^(n) then show : C_(1)-2C_(2)+3C_(3)-....+(-1)^(n-1)n.C_(n)=0

CENGAGE PUBLICATION-BINOMIAL THEOREM-Single Correct Answer
  1. Find underset(r=0) overset(10)sumr^ (10)C(r).3^(r).(-2)^(10-r)

    Text Solution

    |

  2. a. 15 b. -15 c. 0 d. 51

    Text Solution

    |

  3. If (1+x)^n=C0+C1x+C2x^2+...+Cn x^n , then C0C2+C1C3+C2C4+...+C(n-2)Cn=...

    Text Solution

    |

  4. Text Solution

    |

  5. The value of sum(r=0)^(40) r""^(40)C(r)""^(30)C(r) is (a) 40.^(69)C(29...

    Text Solution

    |

  6. The value of sum(r=1)^(15)(r2^r)/((r+2)!) is (a).((17)!-2^16)/((17)!) ...

    Text Solution

    |

  7. (n+2)C0(2^(n+1))-(n+1)C1(2^(n))+(n)C2(2^(n-1))-.... is equal to

    Text Solution

    |

  8. The value of sum(r=0)^(20)(-1)^(r )(""^(50)C(r))/(r+2) is equal to

    Text Solution

    |

  9. sum(r=0)^(300)arx^r=(1+x+x^2+x^3)^(100). If a=sum(r=0)^(300)ar, then s...

    Text Solution

    |

  10. sum(r=0)^(300)arx^r=(1+x+x^2+x^3)^(100). If a=sum(r=0)^(300)ar, then s...

    Text Solution

    |

  11. The value of sum(r=0)^(20)r(20-r)(^(20)Cr)^2 is equal to ?

    Text Solution

    |

  12. If f(x) = ""^(40)C(1).x(1-x)^(39) + 2.""^(40)C(2).x^(2)(1-x)^(38)+3.""...

    Text Solution

    |

  13. Find the value of (sumsum)(0leiltjlen) (i+j)(""^(n)C(i)+""^(n)C(j)).

    Text Solution

    |

  14. In the expansion of [(1+x)//(1-x)]^2, the coefficient of x^n will be a...

    Text Solution

    |

  15. The sum of 1+n(1-1/x)+(n(n+1))/(2!)(1-1/x)^2+.....oo

    Text Solution

    |

  16. sum(k=1)^ook(1-1/n)^(k-1) =?

    Text Solution

    |

  17. The coefficient of x in the expansion of {sqrt(1+x^2)-x}^(-1) in ascen...

    Text Solution

    |

  18. 1+1/3x+(1xx4)/(3xx6)x^2+(1xx4xx7)/(3xx6xx9)x^3+------ is equal to

    Text Solution

    |

  19. 1+1/4 + (1xx3)/(4xx8) + (1xx3xx5)/(4xx8xx12) + "….." is equal to

    Text Solution

    |

  20. If |x|<1,then 1+n((2x)/(1+x))+(n(n+1))/(2!)((2x)/(1+x))^2+...... is eq...

    Text Solution

    |