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The combinatorial coefficient C(n, r) is...

The combinatorial coefficient C(n, r) is equal to

A

number of possible subsets of r members from a set of n distinct members

B

number of possible binary messages of length n with exactly r 1's.

C

number of non decreasing 2-D paths from the lattice point (0, 0) to (r, n).

D

number of ways of selecting 'r' things out of 'n' different things when a particular thing is always included plus the number of ways of selecting ‘r’ things out of 'n', when a particular thing is always excluded.

Text Solution

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The correct Answer is:
A, B, D
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The combinatorial coefficient C(n, r) can not be equal to the (A) number of possible subsets of r members from a set of n distinct members. (B) number of possible binary messages of length n with exactly r 1's. (C) number of non decreasing 2-D paths from the lattice point (0,0) to (r, n) (D) number selecting r things out of n different things when a particular thing is always included plus of ways of the number of ways of selecting r things out of n, when a particular thing is always excluded.

Statement-1 : The sum of the series ^nC_0. ^mC_r+^nC_1.^mC_(r-1)+^nC_2.^mC_(r-2)+......+^nC_r.^mC_0 is equal to ^(n+m)C_r, where C's and C's denotes the combinatorial coefficients in the expansion of (1 + x)^n and (1 + x)^m respectively, Statement-2: Number of ways in which r children can be selected out of (n + m) children consisting of n boys and m girls if each selection may consist of any number of boys and girls is equal to ^(n+m)C_r

Knowledge Check

  • The combinatorial coefficients ""^(n – 1)C_(p) denotes

    A
    The number of ways in which 'n' things of which 'p' are alike and rest different can be arranged in a circle
    B
    The number of ways in which 'p' different things can be selected out of 'n' different thing if a particular thing is always excluded
    C
    Number of ways in which n alike balls can be distributed in 'p' different boxes so that no box remains empty and each box can hold any number of balls.
    D
    The number of ways in which (n – 2) white balls and p black balls can be arranged in a line if black balls are separated, balls are all alike except for the colour.

  • ""^(n) "C"_("r") is equal to-

    A
    `(lfloorn)/( lfloorn-r)`
    B
    `(lfloorn)/( lfloorn+r)`
    C
    `(lfloorn)/( lfloorn lfloorn-r)`
    D
    None of these

  • If ""(n)C_(0), ""(n)C_(1), ""(n)C_(2), ...., ""(n)C_(n), denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q =1 sum_(r=0)^(n) r^(2 " "^n)C_(r) p^(r) q^(n-r) = .

    A
    npq
    B
    np (p+q)
    C
    `np (np + q)`
    D
    ` np (p + nq)`

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    If C_(o)C_(1),C_(2),......,C_(n) denote the binomial coefficients in the expansion of (1+x)^(n), then the value of sum_(r=0)^(n)(r+1)C_(r) is

    If C_(0), C_(1), C_(2),…, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(r=0)^(n)sum_(s=0)^(n)(C_(r) +C_(s))

    If C_(0), C_(1), C_(2), …, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(r=0)^(n)sum_(s=0)^(n)C_(r)C_(s) =

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