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Let .^(n)P(r) denote the number of per...

Let `.^(n)P_(r)` denote the number of permutations of n different things taken r at a time . Then , prove that `1+1.^(1)P_(1)+2.^(2)P_(2)+3.^(3)P_(3)+...+n.^(n)P_(n)=.^(n+1)P_(n+1)`.

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Prove that .^(n)P_(r)=.^(n-1)P_(r)+r.^(n-1)P_(r-1) .

If the number of permutations of n different things taken r at time be denoted by .^(n)P_(r) , show that , (.^(n)P_(1))/(1!)+(.^(n)P_(2))/(2!)+(.^(n)P_(3))/(3!)+...+(.^(n)P_(n))/(n!)=2^(n)-1

Knowledge Check

  • If ""(n)C_(r) denotes the number of combinations of n different things taken r at a time, then the vlaue of ""^(n)C_(r+1)+""^(n)C_(r-1) + 2. ""^(n) C_(r) is-

    A
    `""^(n+2) C_(r+1)`
    B
    `""^(n+1) C_(r+1)`
    C
    `""^(n+2)C_(r)`
    D
    `""^(n+1)C_(r)`

  • The number of combinations of n different of n different things taken r at a time in which p particular things never occur is -

    A
    `.^(n-p)C_(r-p)`
    B
    `.^(n-p)C_(r)`
    C
    `.^(n)C_(r)`
    D
    `((n-p)!)/(r!)`

    • Similar Questions

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    Show that 1+^1P_1+2.^2P_2+3.^3P_3+....+n.^nP_n=^(n+1)P_(n+1) .

    If .^(n)P_(5)=20.^(n)P_(3) , find n.

    (n)p_(r)=k^(n)C_(n-r),k=

    Prove that .^nP_r = ^(n-1)P_r + r^(n-1)P_(r-1)

    If .^(n+1)P_(3)=10.^(n-1)P_(2) , find n.

    If .^(2n+1)P_(n-1): .^(2n-1)P_(n)=3:5 , find n.

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