ΥڡϤƤʥ֥åޡɲΥڡޤϤƤʥ֥åޡ Υڡlivedoor åפɲΥڡޤlivedoor å

[]Ϣ³뼫rĤѤɬr!dzڤ롣

[]Ϣ³뼫rĤѤϡ礭񤱤мΤ褦ˤȽ񤱤롣

n(n-1)(n-2)(n-r+1)(*)

nrǤ롣

r!dzä̤ϡ̤Ǥ롣

\frac{n(n-1)(n-2)~\cdots~(n-r+1)}{r!}(**)
=\frac{n!}{r!~(n-r)!}
=~\,~_n~C_r

nCrnĤΤΤrļФȤ߹碌ʤΤǡǤ롣

Ĥޤꡢ(*)rĤѤr!dzڤ롣

[̾]Ȥ߹碌Ȥ̣դȤʤ

Ǥդǿp(**)ʬn!˴ޤޤĿ(n)pȡʬr!(n-r)!˴ޤޤĿ(r)p+(n-r)pǤ롣

٤ơɤpФƾ˼ΩĤȤ򤤤Ф褤

(n)_p~\ge~(r)_p~+~(n-r)_p(***)

ǡ[]n!˴ޤޤǿpĿ(n)pȤȡΩġ(n)_p~=~[~\frac{n}{p}~]~+~[~\frac{n}{p^2}~]~+~[~\frac{n}{p^3}~]~+~\cdots~=~\Sigma_{k=1}^{\infty}~[\frac{n}{p^k}~]פꡢΩġ

  • (n)_p~=~[~\frac{n}{p}~]~+~[~\frac{n}{p^2}~]~+~[~\frac{n}{p^3}~]~+~\cdots
  • (r)_p~=~[~\frac{r}{p}~]~+~[~\frac{r}{p^2}~]~+~[~\frac{r}{p^3}~]~+~\cdots
  • (r')_p~=~[~\frac{r'}{p}~]~+~[~\frac{r'}{p^2}~]~+~[~\frac{r'}{p^3}~]~+~\cdots

r'=n-r¨n=r+r'

ǡ[][~\frac{a+b}{k}~]~\ge~\frac{a}{k}~+~\frac{b}{k}פꡢ(***)Ωġ

ʸ

  • ؿإݥн10