Home
Class 12
MATHS
Prove that ((n),(r))+2((n),(r-1))+((n)...

Prove that
`((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))`

Text Solution

Verified by Experts

`therefore((n),(r))=.^(n)C_(r)`
`thereforeLHS=((n),(r))+2((n),(r-1))+((n),(r-2))`
`=.^(n)C_(r)+2.^(n)C_(r-1)+.^(n)C_(r-2)`
`=(.^(n)C_(r)+.^(n)C_(r-1))+(.^(n)C_(r-1)+.^(n)C_(r-2))`
`=.^(n+1)C_(r)+.^(n+1)C_(r-1)=.^(n+2)C_(r)`
`=((n+2),(r))=RHS`
Promotional Banner

Topper's Solved these Questions

  • PERMUTATIONS AND COMBINATIONS

    ARIHANT MATHS|Exercise JEE Type Solved Examples: Single Matching Type Questions|1 Videos

  • PERMUTATIONS AND COMBINATIONS

    ARIHANT MATHS|Exercise Exercise For Session 1|11 Videos

  • PARABOLA

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|35 Videos

  • PROBABILITY

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|55 Videos

Similar Questions

Explore conceptually related problems

Prove that "^nC_r+2 ^(n)C_(r-1)+ ^(n)C_(r-2) = ^(n+2)C_r .

Prove that (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!)= ((n+1)!)/ (r!(n-r+1)!) .

Prove that: sum_(r=0)^(n) 3^( r) ""^(n)C_(r) = 4^(n) .

Prove that : ^nP_r= "^(n-1)P_r+r ^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

If t_r is the rth term is the expansion of (1 + a)^n , in ascending powers of a, prove that : r(r+1) t_(r+2)= (n-r+1)(n-r) a^2 t_r .

Prove that underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0 , where k = (3n)/2 and n is an even integer.

If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

Statement-1: sum_(r =0)^(n) (r +1)""^(n)C_(r) = (n +2) 2^(n-1) Statement -2: sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) = (1 + x)^(n) + nx (1 + x)^(n-1)

the value of overset(n)underset(r=2)(Simga)(-2)^(r ) |{:( .^(n-2)C_(r-2),,.^(n-2)C_(r-1),,.^(n-2)C_(r)),(-3,,1 ,,1),(2,,-1,,0):}| (n gt 2)

If ([""^(n)C_(r) + 4*""^(n)C_(r+1) + 6*""^(n)C_(r+2)+ 4*""^(n)C_(r+3) + ""^(n)C_(r+4)])/([""^(n)C_(r) + 3. ""^(n)C_(r+1)+ 3*""^(n)C_(r+2) + ""^(n)C_(r +3)])=(n + lambda)/(r+lambda) the value of lambda is

ARIHANT MATHS-PERMUTATIONS AND COMBINATIONS -Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that ((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))

    Text Solution

    |

  2. There is a rectangular sheet of dimension (2m-1)xx(2n-1), (where m > 0...

    Text Solution

    |

  3. If the letters of the word SACHIN are arranged in all possible ways ...

    Text Solution

    |

  4. lf r, s, t are prime numbers and p, q are the positive integers such t...

    Text Solution

    |

  5. At an election, a voter may vote for any number of candidates not grea...

    Text Solution

    |

  6. The letters of the word COCHIN are permuted and all the permutation...

    Text Solution

    |

  7. The set S""=""{1,""2,""3,""........ ,""12) is to be partitioned into...

    Text Solution

    |

  8. Consider all possible permutations of the letters of the word ENDEANOE...

    Text Solution

    |

  9. The number of words can be formed with the letters of the word 'MATHEM...

    Text Solution

    |

  10. In a shop, there are five types of ice-creams available. A child buys ...

    Text Solution

    |

  11. The number of seven digit integers, with sum of the digits equal to 10...

    Text Solution

    |

  12. Three boys and two girls stand in a queue. The probability that the nu...

    Text Solution

    |

  13. There are two urns. Urn A has 3 distinct red balls and urn B has 9 ...

    Text Solution

    |

  14. In how many ways 5 different balls can be distributed into 3 boxes so ...

    Text Solution

    |

  15. There are 10 points in a plane, out of these 6 are collinear. The numb...

    Text Solution

    |

  16. The total number of ways in which 5 balls of differert colours can be ...

    Text Solution

    |

  17. Let n denote the number of all n-digit positive integers formed by the...

    Text Solution

    |

  18. Let a(n) denote the number of all n-digit numbers formed by the digits...

    Text Solution

    |

  19. Assuming the balls to be identical except for difference in colours, t...

    Text Solution

    |

  20. Let Tn be the number of all possible triangles formed by joining ve...

    Text Solution

    |

  21. Consider the set of eight vector V={a hat i+b hat j+c hat k ; a ,bc in...

    Text Solution

    |