❈❍❆P❚❊❘ ✶
❈♦♠❜✐♥❛t♦r✐❛❧ ❆♥❛❧②s✐s
✶✳✶✳ ❚❤❡ ❇❛s✐❝ P✐♥❝✐♣❧❡ ♦❢ ❈♦✉♥t✐♥❣
❆ ❝♦♠♠✉♥✐❝❛t✐♦♥ s②st❡♠ ✐s t♦ ❝♦♥s✐st ♦❢ ♥ s❡❡♠✐♥❣❧② ✐❞❡♥t✐❝❛❧ ❛♥t❡♥♥❛s t❤❛t ❛r❡ t♦ ❜❡ ❧✐♥❡❞ ✉♣ ✐♥ ❛ ❧✐♥❡❛r ♦r❞❡r✳ ❚❤❡ r❡s✉❧t✐♥❣ s②st❡♠ ✇✐❧❧ t❤❡♥ ❜❡ ❛❜❧❡ t♦ r❡❝❡✐✈❡ ❛❧❧ ✐♥❝♦♠✐♥❣ s✐❣♥❛❧s✖❛♥❞ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ ❢✉♥❝t✐♦♥❛❧✖❛s ❧♦♥❣ ❛s ♥♦ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❛♥t❡♥♥❛s ❛r❡ ❞❡❢❡❝t✐✈❡✳ ■❢ ✐t t✉r♥s ♦✉t t❤❛t ❡①❛❝t❧② ♠ ♦❢ t❤❡ ♥ ❛♥t❡♥♥❛s ❛r❡ ❞❡❢❡❝t✐✈❡✱ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ r❡s✉❧t✐♥❣ s②st❡♠ ✇✐❧❧ ❜❡ ❢✉♥❝t✐♦♥❛❧❄ ❋♦r ✐♥st❛♥❝❡✱ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡r❡ ♥ ❂ ✹ ❛♥❞ ♠ ❂ ✷✱ t❤❡r❡ ❛r❡ ✻ ♣♦ss✐❜❧❡ s②st❡♠ ❝♦♥✜❣✉r❛t✐♦♥s✱ ♥❛♠❡❧②✱
❈♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ✐♥ ✇❤✐❝❤ ❛♥ ❡✈❡♥t ♠❛② ♦❝❝✉r ❝❛♥ ❜❡ ❛ t❡❞✐♦✉s ♣♦r❜❧❡♠ ✐♥ ❝♦♠♣❧✐❝❛t❡❞ ❡①♣❡r✐♠❡♥t✳ ❆ ❢❡✇ ❤❡❧♣❢✉❧ ❝♦✉♥t✐♥❣ t❡❝❤♥✐q✉❡ ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞✳
❊①❛♠♣❧❡ ✶✳ ❍♦✇ ♠❛♥② ✇❛②s ❝❛♥ ❛ ✷✵✲q✉❡st✐♦♥s tr✉❡✲❢❛❧s❡ ❜❡ ❛♥s✇❡r❡❞❄ ❙♦❧✉t✐♦♥✳
❊①❛♠♣❧❡ ✷✳ ✭❛✮ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ✼✲♣❧❛❝❡ ❧✐❝❡♥s❡ ♣❧❛t❡s ❛r❡ ♣♦ss✐❜❧❡ ✐❢ t❤❡ ✜rst ✸ ♣❧❛❝❡s ❛r❡ t♦ ❜❡ ♦❝❝✉♣✐❡❞ ❜② ❧❡tt❡rs ❛♥❞ t❤❡ ✜♥❛❧ ✹ ❜② ♥✉♠❜❡rs❄ ✭❜✮ ❆♥❞ ❤♦✇ ♠❛♥② ❧✐❝❡♥s❡ ♣❧❛t❡s ✇♦✉❧❞ ❜❡ ♣♦ss✐❜❧❡ ✐❢ r❡♣❡t✐t✐♦♥ ❛♠♦♥❣ ❧❡tt❡rs ♦r ♥✉♠❜❡rs ✇❡r❡ ♣r♦❤✐❜✐t❡❞❄
❙♦❧✉t✐♦♥✳
❚❤❡♦r❡♠ ✸✳ ■❢ t❤❡ ❛r❡N ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ♦❢ ❡❛❝❤ ♦❢ r tr✐❛❧s ♦❢ ❛♥ ❡①♣❡r✐♠❡♥t✱ t❤❡♥ t❤❡r❡ ❛r❡ Nr ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ✐♥ t❤❡ s❛♠♣❧❡ s♣❛❝❡✳
✶✳✷✳ P❡r♠✉t❛t✐♦♥s
❆♥ ♦r❞❡r❡❞ ❛rr❛♥❣❡♠❡♥t ♦❢ ❛ s❡t ♦❢ ♦❜❥❡❝ts ✐s ❦♥♦✇♥ ❛s ❛ ♣❡r♠✉t❛t✐♦♥✳ ❊①❛♠♣❧❡ ✹✳ ❍♦✇ ♠❛♥② ♥✉♠❜❡r ♦❢ ❛rr❛♥❣❡♠❡♥ts ❛r❡ ♣♦ss✐❜❧❡ ❢♦r t❤r❡❡ ❞✐st✐♥❝t ❧❡tt❡rs ❆✱ ❇✱ ❈ ❄
❙♦❧✉t✐♦♥✳
❚❤❡♦r❡♠ ✺✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢n ❞✐st✐♥❣✉✐s❤❛❜❧❡ ♦❜❥❡❝ts ✐sn!
Pr♦♦❢✳ ❚❤✐s ❢♦❧❧♦✇s ❜② ❛♣♣❧②✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♣r✐♥❝✐♣❧❡✳ ❚♦ ✜❧❧ n ♣♦s✐✲ t✐♦♥s ✇✐t❤n❞✐st✐♥❝t ♦❜❥❡❝ts✱ t❤❡ ✜rst ♣♦s✐t✐♦♥ ♠❛② ❜❡ ✜❧❧❡❞n✇❛②s ✉s✐♥❣ ❛♥② ♦♥❡ ♦❢ t❤❡n♦❜❥❡❝ts✱ t❤❡ s❡❝♦♥❞ ♣♦s✐t✐♦♥ ♠❛② ❜❡ ✜❧❧❡❞n−1✇❛②s ✉s✐♥❣ ❛♥② r❡♠❛✐♥✐♥❣n−1
♦❜❥❡❝ts✱ ❛♥❞ s♦ ♦♥ ✉♥t✐❧ t❤❡ ❧❛st ♦❜❥❡❝t ✐s ♣❧❛❝❡❞ ✐♥ t❤❡ ❧❛st ♣♦s✐t✐♦♥✳ ❚❤✉s ❜② t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♣r✐♥❝✐♣❧❡✱ t❤✐s ♦♣❡r❛t✐♦♥ ♠❛② ❜❡ ❝❛rr✐❡❞ ♦✉t ✐♥ n.(n−1)...1 = n!
✇❛②s✳
✶✳✷✳ P❊❘▼❯❚❆❚■❖◆❙ ✷
❖♥❡ ❛❧s♦ ♠❛② ❜❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ♦❢ s❡❧❡❝t✐♥❣ r♦❜❥❡❝ts ❢r♦♠ n❞✐st✐♥❝t ♦❜❥❡❝ts ❛♥❞ t❤❡♥ ♦r❞❡r✐♥❣ t❤❡s❡r ♦❜❥❡❝ts✳
❊①❛♠♣❧❡ ✻✳ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❛rr❛♥❣❡♠❡♥ts ❛r❡ ♣♦ss✐❜❧❡ ✐❢ ✜✈❡ ❞✐st✐♥❝t ❧❡tt❡rs ❆✱ ❇✱ ❈✱ ❉✱ ❊ t❛❦❡♥ t❤r❡❡ ❛t ❛ t✐♠❡❄
❙♦❧✉t✐♦♥✳
❚❤❡♦r❡♠ ✼✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢n❞✐st✐♥❝t ♦❜❥❡❝ts t❛❦❡♥r❛t t✐♠❡ ✐s
Pn,r =
n! (n−r)!
❊①❛♠♣❧❡ ✽✳ ■❢ t❤r❡❡ ❞✐st✐♥❝t ❧❡tt❡rs ❆✱ ❇✱ ❈ ❛r❡ r❛♥❞♦♠❧② ❛rr❛♥❣❡❞ ✐♥ ❛ ❝✐r❝❧❡✱ ❤♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❛rr❛♥❣❡♠❡♥ts ❛r❡ ♣♦ss✐❜❧❡❄
❙♦❧✉t✐♦♥✳
❚❤❡♦r❡♠ ✾✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢ n ❞✐st✐♥❝t ♦❜❥❡❝ts ❛rr❛♥❣❡❞ ✐♥ ❛ ❝✐r❝❧❡ ✐s
(n−1)!
❊①❛♠♣❧❡ ✶✵✳ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❧❡tt❡r ❛rr❛♥❣❡♠❡♥ts ❝❛♥ ❜❡ ❢♦r♠❡❞ ❢r♦♠ t❤❡ ❧❡tt❡rsEY E❄
❙♦❧✉t✐♦♥✳
❚❤❡ ❝♦♥❝❡♣t ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐③❡❞ t♦ t❤❡ ❝❛s❡ ♦❢ ♣❡r♠✉t✐♥❣kt②♣❡s ♦❢ ♦❜❥❡❝ts✳ ❚❤❡♦r❡♠ ✶✶✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢n♦❜❥❡❝ts ♦❢ ✇❤✐❝❤r1 ❛r❡ ♦❢ ♦♥❡
❦✐♥❞✱r2 ❛r❡ ♦❢ ❛ s❡❝♦♥❞ ❦✐♥❞✱ ✳✳✳✱rk ♦❢ ❛kt❤ ❦✐♥❞ ✐s
n!
r1!r2!...rk!
❊①❛♠♣❧❡ ✶✷✳ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❧❡tt❡r ❛rr❛♥❣❡♠❡♥ts ❝❛♥ ❜❡ ❢♦r♠❡❞ ❢r♦♠ t❤❡ ❧❡tt❡rsP EP P ER❄
❙♦❧✉t✐♦♥✳
❊①❛♠♣❧❡ ✶✸✳ ■❢ ✹ ❞✐✛❡r❡♥t ♣♦♣s✐❝❧❡s ❛r❡ t♦ ❜❡ ❞✐str✐❜✉t❡❞ ❡q✉❛❧❧② ❛♠♦♥❣ t✇♦ ❝❤✐❧❞r❡♥✱ ❤♦✇ ♠❛♥② ❞✐✈✐s✐♦♥s ❛r❡ ♣♦ss✐❜❧❡❄
❙♦❧✉t✐♦♥✳
❚❤❡♦r❡♠ ✶✹✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ♦❢ ♣❛rt✐t✐♦♥✐♥❣ ❛ s❡t ♦❢ n ♦❜❥❡❝ts ✐♥t♦ k ❝❡❧❧s ✇✐t❤r1 ♦❜❥❡❝ts ✐♥ t❤❡ ✜rst ❝❡❧❧✱r2 ♦❜❥❡❝ts ✐♥ t❤❡ s❡❝♦♥❞ ❝❡❧❧✱ ❛♥❞ s♦ ❢♦rt❤ ✐s
n r1, r2, ...rk
= n!
r1!r2!...rk!
✇❤❡r❡r1+r2+...+rk =n✳
P❛rt✐t✐♦♥✐♥❣ ❛ss✉♠❡s t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts t♦ ❜❡ ♣❧❛❝❡❞ ✐♥ ❡❛❝❤ ❝❡❧❧ ✐s ✜①❡❞✱ ❛♥❞ t❤❛t t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ♦❜❥❡❝t ❛r❡ ♣❧❛❝❡❞ ✐♥t♦ ❝❡❧❧s ✐s ♥♦t ❝♦♥s✐❞❡r❡❞✳ ❊①❛♠♣❧❡ ✶✺✳ ■❢ ✶✷ ♣❡♦♣❧❡ ❛r❡ t♦ ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ ✸ ❝♦♠♠✐tt❡❡s ♦❢ r❡s♣❡❝t✐✈❡ s✐③❡s ✸✱ ✹✱ ❛♥❞ ✺✱ ❤♦✇ ♠❛♥② ❞✐✈✐s✐♦♥s ❛r❡ ♣♦ss✐❜❧❡❄
✶✳✹✳ P❘❖❇▲❊▼❙ ✺
✭❜✮ ❆♥s✇❡r ✭❛✮ ✐❢ ❡❛❝❤ ♣♦s✐t✐♦♥ ✐s ✉s❡❞ t❤r❡❡ t✐♠❡s
✭✶✻✮ ❙✉♣♣♦s❡ t❤❡ ✇✐♥♥✐♥❣ ♥✉♠❜❡r ✐♥ ❛ ❧♦tt❡r② ✐s ❛ ❢♦✉r✲❞✐❣✐t ♥✉♠❜❡r ❞❡t❡r♠✐♥❡❞ ❜② ❞r❛✇✐♥❣ ❢♦✉r s❧✐♣s ♦❢ ♣❛♣❡r ✭✇✐t❤♦✉t r❡♣❧❛❝❡♠❡♥t✮ ❢r♦♠ ❛ ❜♦① t❤❛t ❝♦♥t❛✐♥s ♥✐♥❡ s❧✐♣s ♥✉♠❜❡r❡❞ ❝♦♥s❡❝✉t✐✈❡❧② ✶ t❤r♦✉❣❤ ✾ ❛♥❞ t❤❡♥ r❡❝♦r❞✐♥❣ t❤❡ ❞✐❣✐ts ✐♥ ♦r❞❡r ❢r♦♠ t❤❡ s♠❛❧❧❡st t♦ ❧❛r❣❡st✳
✭❛✮ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❧♦tt❡r② ♥✉♠❜❡rs ❛r❡ ♣♦ss✐❜❧❡❄
