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Math Help - nPr- arrangements and permutations

  1. #1
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    nPr- arrangements and permutations

    Prove from the defintion of nPr (as in nPr = n!/(n-r)!) that
    (n+1)/Pr = nPr + r.nP(r-1)
    and show that nPr = (n-2)Pr + 2r. (n-2)P(r-1) + r(r-1) x (n-2)P(r-2)

    if anyone could give me some techniques or some help solving these types of questions i would really appreciate it ! thankyou =D
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by flyinhigh123 View Post
    Prove from the defintion of nPr (as in nPr = n!/(n-r)!) that
    (n+1)/Pr = nPr + r.nP(r-1)
    and show that nPr = (n-2)Pr + 2r. (n-2)P(r-1) + r(r-1) x (n-2)P(r-2)

    if anyone could give me some techniques or some help solving these types of questions i would really appreciate it ! thankyou =D
    When trying to prove equality between two expressions we take one side of the statement and apply 'known' transformations to see if we can get to equal the other side. Here is some help with the first part. Let me know if the algebra is ok:

    Right side =
    nPr + r  nP(r-1) = \frac{n!}{(n-r)!}  + r \frac{n!}{(n-(r-1))!}=
    =\frac{n!}{(n-r)!}  + r \frac{n!}{(n-r+1)!}=\frac{n!}{(n-r)!}  + r \frac{n!}{(n-r+1)(n-r)!}=
    \frac{n! (n-r+1)}{(n-r+1)(n-r)!}  + r \frac{n!}{(n-r+1)(n-r)!}= \frac{n! (n-r+1)}{(n-r+1)!}  +  \frac{r*n!}{(n-r+1)!}=
    \frac{n! (n-r+1+r)}{(n-r+1)!}= \frac{n! (n+1)}{([n+1]-r)!} = \frac{[n+1]!}{([n+1]-r)!} = (n+1)Pr = Left Hand Side

    Good luck!!
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