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Math Help - arrangements help

  1. #1
    Junior Member
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    arrangements help

    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
    MHF Contributor red_dog's Avatar
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    P_n^r+rP_n^{r-1}=\frac{n!}{(n-r)!}+r\cdot\frac{n!}{(n-r+1)!}=

    =\frac{n!(n-r+1)}{(n-r+1)!}+\frac{n!r}{(n-r+1)!}=\frac{n!(n-r+1+r)}{(n-r+1)!}=\frac{(n+1)!}{(n-r+1)!}=P_{n+1}^r

    Now we use this formula to prove the second identity.

    P_n^r=P_{n-1}^r+rP_{n-1}^{r-1}=P_{n-2}^r+rP_{n-2}^{r-1}+r(P_{n-2}^{r-1}+(r-1)P_{n-2}^{r-2})=

    =P_{n-2}^r+2rP_{n-2}^{r-1}+r(r-1)P_{n-2}^{r-2}
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  3. #3
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    thanx so much for your help !
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