Open Probabilidade aplicada aos jogos de azar

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

PP●▼ ✲ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❆ Pr♦❜❛❜✐❧✐❞❛❞❡ ❆♣❧✐❝❛❞❛ ❛♦s

❏♦❣♦s ❞❡ ❆③❛r

♣♦r

❘❛❢❛❡❧ ❚❤é ❇♦♥✐❢á❝✐♦ ❞❡ ❆♥❞r❛❞❡

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❆❧❡①❛♥❞r❡ ❞❡ ❇✉st❛♠❛♥t❡ ❙✐♠❛s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

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A553p Andrade, Rafael Thé Bonifácio de.

A probabilidade aplicada aos jogos de azar / Rafael Thé Bonifácio de Andrade.- João Pessoa, 2017.

69f. : il.

Orientador: Alexandre de Bustamante Simas Dissertação (Mestrado) - UFPB/CCEN

1. Matemática. 2. Teoria dos jogos. 3. Probabilidade. 4. Jogos de azar.

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A Probabilidade Aplicada aos

Jogos de Azar

por

Rafael Thé Bonifácio de Andrade

Dissertação apresentada ao Corpo Docente do Mestrado Profissional em Matemática em Rede Nacional PROFMAT CCEN-UFPB, como requisito parcial para obtenção do título de Mestre em Matemática.

Área de Concentração: probabilidade

Aprovada por:

Prof.

r. Henrique de Barros Correia Vitório - UFPE

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❆❣r❛❞❡❝✐♠❡♥t♦s

◆❡st❡ ♠♦♠❡♥t♦ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ✐♠❡♥s❛♠❡♥t❡ ❛ ✈ár✐❛s ♣❡ss♦❛s q✉❡ ❝♦♥tr✐✲ ❜✉ír❛♠ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❡ss❡ ✐♠♣♦rt❛♥t❡ ♣❛ss♦✱ ♠❛s ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣❡❧❛ ✈✐❞❛ ❡ s❛ú❞❡ ❢ís✐❝❛ ❡ ♠❡♥t❛❧ ♣❛r❛ ♣❡r❝♦rr❡r ❡st❡ ❧♦♥❣♦ ❝❛♠✐♥❤♦ ❛té ❛q✉✐✳

●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ♠❡✉s ❛✈ós ❉✐♥❛rt❡ ❡ ❖♥❡✐❞❛ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❞❛❞♦ ❛♦ ♣♦♥t❛♣é ✐♥✐❝✐❛❧ ❡ s❡❣✉r❛r❡♠ ❛ ❜❛rr❛ ❡♠ ✈ár✐♦s ♠♦♠❡♥t♦s ❞❡ ❢r❛q✉❡③❛ ❡ ♣r❡❝✐sã♦✳

❆♦s ♠❡✉s ♣❛✐s ❆♥❛ ❚❡r❡③❛ ❡ ◆❡❧s♦♥ ❘✐❝❛r❞♦ ✭✐♥ ♠❡♠♦r✐❛♥✮ ♣❡❧❛ ❢♦r♠❛çã♦ ♣❡ss♦❛❧ ❡ ✐♥t❡❧❡❝t✉❛❧ ❞❡s❞❡ ♠❡✉ ♥❛s❝✐♠❡♥t♦ ❛té ❤♦❥❡✱ ♦s q✉❛✐s ❢♦r❛♠ ❡①❡♠♣❧♦ ❞❡ ❝❛rát❡r✱ ♣r♦✜ss✐♦♥❛❧✐s♠♦ ❡ ❧✉t❛ ♣❡❧❛ s♦❜r❡✈✐✈ê♥❝✐❛✳

❆♦s ♠❡✉s ♣❛❞r✐♥❤♦s ▼❛r✐❛ ❞❡ ❋át✐♠❛ ❆♥❞r❛❞❡ ▼❡❧♦ ❡ ■❧s♦♥ ❞❡ ▼❡❧♦ ❋✐❧❤♦ ♣❡❧♦ ❛❝♦♠♣❛♥❤❛♠❡♥t♦ ❡ ❝♦❜r❛♥ç❛s ❞❛ ♠✐♥❤❛ ❡❞✉❝❛çã♦ ❡ ♠❡✉ ❧❛❞♦ ♣r♦✜ss✐♦♥❛❧✳

❆ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s ❞❛ t✉r♠❛ ✷✵✶✺ ❞♦ P❘❖❋▼❆❚ q✉❡ ❡st✐✈❡r❛♠ ❝♦♠✐❣♦ ❞❡s❞❡ ♦ ✐♥í❝✐♦✱ ♠❡ ❛♣♦✐❛♥❞♦ ❡ ♠❡ ❛❝♦♥s❡❧❤❛♥❞♦✱ ♦s q✉❛✐s ❡✉ ♥✉♥❝❛ ❡sq✉❡❝❡r❡✐✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♣❡ss♦❛ ❞❡ ▼❛✐❧s♦♥ ❆❧✈❡s q✉❡ s❡ t♦r♥♦✉ ✉♠ ❞♦s ♠❡❧❤♦r❡s ❛♠✐❣♦s q✉❡ ❉❡✉s ❝♦❧♦❝♦✉ ♥♦ ♠❡✉ ❝❛♠✐♥❤♦✳

❆✐♥❞❛ ♥♦ â♠❜✐t♦ ❞♦ P❘❖❋▼❆❚✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦ ♣r♦❢✳ ❉r✳ ❇r✉♥♦ ❘✐❜❡✐r♦ ♣♦r t♦❞❛ ♣❛❝✐ê♥❝✐❛ ❡ ♦r✐❡♥t❛çõ❡s ❛♦ ❧♦♥❣♦ ❞♦ ♣r♦❣r❛♠❛✱ ❡ ❛♦ ♣r♦❢✳ ❉r✳ ❆❧❡①❛♥❞r❡ ❞❡ ❇✉st❛♠❛♥t❡s ❙✐♠❛s✱ ♦ q✉❛❧ ♠❡ ♦r✐❡♥t♦✉ ♠❡s♠♦ ❢r❡♥t❡ ❛ s✐t✉❛çõ❡s ♣❡ss♦❛✐s ❛❞✈❡rs❛s✳

❊ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ♠✐♥❤❛ ❡s♣♦s❛ ❚❤❛ís❛ ❆♥❞r❛❞❡ ♣♦r t♦❞❛ ❢♦rç❛✱ ❝♦♠♣r❡❡♥sã♦✱ ❛♠♦r✱ ❛❥✉❞❛ ❡ t❛♥t❛s ♦✉tr❛s ✈✐rt✉❞❡s q✉❡ ❞❡♠♦♥str♦✉ t❡r ❝♦♠✐❣♦ ❞✉r❛♥t❡ t♦❞♦ t❡♠♣♦ q✉❡ ♥♦s ❝♦♥❤❡❝❡♠♦s✱ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❡st❛ é♣♦❝❛✱ ✜❝❛♥❞♦ ❛♦ ♠❡✉ ❧❛❞♦ ♣❛r❛ ♦ q✉❡ q✉❡r q✉❡ ❛❝♦♥t❡❝❡ss❡✱ ❛ q✉❛❧ ♥ã♦ t❡r✐❛ ❝♦♠♦ ❡①♣r❡ss❛r ♥❡♠ ♠❡♥s✉r❛r ♦ t❛♠❛♥❤♦ ❞♦ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡ ♦ q✉❡ s✐♥t♦ ♣♦r ❡❧❛✳

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❉❡❞✐❝❛tór✐❛

❆ ❉❡✉s✳ ❆ ❚❤❛ís❛✳ ❆♦s ♠❡✉s

❢❛♠✐❧✐❛r❡s✳ ➚q✉❡❧❡s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ♥❡st❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆ t♦❞♦s ♦s q✉❡ s❡ ❛❧❡❣r❛♠ ❝♦♠ ♠✐♥❤❛ ✈✐tór✐❛✳

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❘❡s✉♠♦

❖s ❥♦❣♦s sã♦ ♣r❡s❡♥t❡s ❡♠ t♦❞❛s ❛s ❢❛s❡s ❞❛ ✈✐❞❛ ❞♦ s❡r ❤✉♠❛♥♦ ❡ ❛❧❣✉♥s ❞❡❧❡s sã♦ ❝♦♥s✐❞❡r❛❞♦s ❏♦❣♦s ❞❡ ❆③❛r✳ ❆ t❡♦r✐❛ ❞♦s ❥♦❣♦s é ♦ r❛♠♦ ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ ❡st✉❞❛ ♠♦❞❡❧♦s ❞❡ ❞❡❝✐sã♦ ♦♥❞❡ ♦ ♦❜❥❡t✐✈♦ é t❡r ❣❛♥❤♦s✱ ❡ é ❛♣❧✐❝á✈❡❧ ❛ ❞✐✈❡rs♦s ❡st✉❞♦s ❝♦♠♣♦rt❛♠❡♥t❛✐s ✐♥❝❧✉✐♥❞♦ ❡❝♦♥♦♠✐❛✱ ❝✐ê♥❝✐❛s ♣♦❧ít✐❝❛s✱ ♣s✐❝♦❧♦❣✐❛ ❡ ❧ó❣✐❝❛✳ ❖s ❥♦❣♦s ❡st✉❞❛❞♦s ♥❡st❛ t❡♦r✐❛ ♣♦ss✉❡♠ ❡❧❡♠❡♥t♦s ❜❡♠ ❞❡✜♥✐❞♦s ❝♦♠♦ ❥♦❣❛❞♦r❡s✱ ✐♥❢♦r♠❛çõ❡s ❡ ❛çõ❡s✳ ◆❡st❡ tr❛❜❛❧❤♦ ✈❡r❡♠♦s q✉❡ ♦s ❏♦❣♦s ❞❡ ❆③❛r sã♦ ❛q✉❡❧❡s q✉❡ t❡♠ ❛ ♠❛✐♦r ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❞❡rr♦t❛ ❞♦ q✉❡ ❞❡ ✈✐tór✐❛✱ tr❛t❛r❡♠♦s ❞❡ ❛❧❣✉♥s ❥♦❣♦s ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞♦s ❡ ❝♦♠✉♥s ❝♦♠♦✿ Pôq✉❡r✱ ❇❧❛❝❦❥❛❝❦✱ ❈r❛♣s✱ ❘♦❧❡t❛ ❡ ▲♦t❡r✐❛ ❝♦♠♦ ❛ ▼❡❣❛✲❙❡♥❛✳ ▼♦str❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ss❡s ❥♦❣♦s✱ ✉♠ ♣♦✉❝♦ ❞❛s s✉❛s ❤✐s✲ tór✐❛s ❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ✉♠ ❥♦❣❛❞♦r ♦❜t❡r s✉❝❡ss♦ ❛♦ ❥♦❣❛r✱ ❛ ✜♠ ❞❡ ❞❡♠♦♥str❛r ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❛s r❡❛✐s ❝❤❛♥❝❡s ❞❡ s❡ ❣❛♥❤❛r ❛♦ ❥♦❣❛r ♦s ❢❛♠♦s♦s ❥♦❣♦s ❞❡ ❛③❛r✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ t❡♦r✐❛ ❞♦s ❥♦❣♦s❀ ♣r♦❜❛❜✐❧✐❞❛❞❡❀ ❥♦❣♦s ❞❡ ❛③❛r✳

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❆❜str❛❝t

●❛♠❡s ❛r❡ ♣r❡s❡♥t ✐♥ ❛❧❧ ♣❤❛s❡s ♦❢ ❤✉♠❛♥ ❧✐❢❡ ❛♥❞ s♦♠❡ ♦❢ t❤❡♠ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❣❛♠❡ ♦❢ ❝❤❛♥❝❡✳ ●❛♠❡ t❤❡♦r② ✐s ❛ ❜r❛♥❝❤ ♦❢ ♠❛t❤❡♠❛t✐❝s ❝♦♥❝❡r♥❡❞ ✐♥ ❞❡❝✐s✐♦♥ ♠♦❞❡❧s ✇❤❡r❡ t❤❡ ❣♦❛❧ ✐s t♦ ❣❛✐♥✱ ❛♥❞ ✐s ❛♣♣❧✐❝❛❜❧❡ t♦ s❡✈❡r❛❧ ❜❡❤❛✈✐♦r❛❧ st✉❞✐❡s ✐♥❝❧✉❞✐♥❣ ❡❝♦♥♦♠✐❝s✱ ♣♦❧✐t✐❝❛❧ s❝✐❡♥❝❡✱ ♣s②❝❤♦❧♦❣②✱ ❛♥❞ ❧♦❣✐❝✳ ❚❤❡ ❣❛♠❡s st✉❞✐❡❞ ✐♥ t❤✐s t❤❡♦r② ❤❛✈❡ ✇❡❧❧ ❞❡✜♥❡❞ ❡❧❡♠❡♥ts s✉❝❤ ❛s ♣❧❛②❡rs✱ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ ❛❝t✐♦♥s✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ s❡❡ t❤❛t ❣❛♠❡s ♦❢ ❝❤❛♥❝❡ ❛r❡ ❣❛♠❡s t❤❛t ❛r❡ ♠♦r❡ ❧✐❦❡❧② t♦ ❜❡ ❞❡❢❡❛t❡❞ t❤❛♥ ✇✐♥✱ ✇❡ ✇✐❧❧ ❞❡❛❧ ✇✐t❤ s♦♠❡ ✇❡❧❧ ❦♥♦✇♥ ❛♥❞ ❝♦♠♠♦♥ ❣❛♠❡s s✉❝❤ ❛s✿ P♦❦❡r✱ ❇❧❛❝❦✲ ❥❛❝❦✱ ❈r❛♣s✱ ❘♦✉❧❡tt❡ ❛♥❞ ▲♦tt❡r② ❛s t❤❡ ▼❡❣❛✲❙❡♥❛✳ ❙❤♦✇ ❤♦✇ t❤❡s❡ ❣❛♠❡s ✇♦r❦✱ t❤❡✐r st♦r✐❡s ❛♥❞ t❤❡ ♦❞❞s ♦❢ ❛ ♣❧❛②❡r t♦ ❜❡ s✉❝❝❡ss❢✉❧ ✐♥ ♣❧❛②✐♥❣✱ ✐♥ ♦r❞❡r t♦ s❤♦✇ ♠❛t❤❡♠❛t✐❝❛❧❧② t❤❡ r❡❛❧ ❝❤❛♥❝❡s ♦❢ ✇✐♥♥✐♥❣ ✇❤❡♥ ♣❧❛②✐♥❣ t❤❡s❡ ❢❛♠♦✉s ❣❛♠❡s✳

❑❡② ✇♦r❞s✿ ❣❛♠❡s t❤❡♦r② ❀ ♣r♦❜❛❜✐❧✐t② ❀ ❣❛♠❜❧✐♥❣✳

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❙✉♠ár✐♦

❆❣r❛❞❡❝✐♠❡♥t♦s ✐✐✐

✶ ❘❡✈✐s❛♥❞♦ ❝♦♥❝❡✐t♦s ✶

✶✳✶ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ Pr✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❝♦♥t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✷ ❋❛t♦r✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✸ P❡r♠✉t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✹ ❆❣r✉♣❛♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✷ ❈❛❧❝✉❧❛♥❞♦ Pr♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷ ❖ q✉❡ é ✉♠ ❥♦❣♦ ❞❡ ❛③❛r ✶✹

✷✳✶ ❖s ❥♦❣♦s ❞❡ ❛③❛r ♥♦ ❇r❛s✐❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✶✳✶ ❏♦❣♦s ♣❡r♠✐t✐❞♦s ♣♦r ❧❡✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✷ ❏♦❣♦s ✐❧í❝✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❆s♣❡❝t♦s ♣s✐❝♦❧ó❣✐❝♦s ❞♦s ❥♦❣♦s ❞❡ ❛③❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸ Pôq✉❡r ✶✽

✸✳✶ ❆ ❤✐stór✐❛ ❞♦ Pôq✉❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷ ❖s ✈ár✐♦s t✐♣♦s ❞❡ ♣ôq✉❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✸ ❖ ❚❡①❛s ❤♦❧❞✬❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸✳✶ ❆♣r❡♥❞❡♥❞♦ ❛ ❥♦❣❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✸✳✷ ❆s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ ♣♦♥t✉❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹ ❇❧❛❝❦❥❛❝❦ ✸✸

✹✳✶ ❆ ❤✐stór✐❛ ❞♦ ❇❧❛❝❦❥❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✷ ❆s r❡❣r❛s ❞♦ ❇❧❛❝❦❥❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✷✳✶ P♦♥t✉❛♥❞♦ ♥♦ ❇❧❛❝❦❥❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✷✳✷ ❏♦❣❛♥❞♦ ♦ ❇❧❛❝❦❥❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✸ ❆s ♣r♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✸✳✶ ✷✶ ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✸✳✷ ❖✉tr❛s ❢♦r♠❛s ❞❡ ♣♦♥t✉❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✸✳✸ P♦♥t♦s ✐♥s✉✜❝✐❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✸✳✹ ❊①tr❛♣♦❧❛çã♦ ❞♦s ✷✶ ♣♦♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✸✳✺ ❱✐❛❜✐❧✐❞❛❞❡ ♣❛r❛ ♦ ❥♦❣❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✹ ❈♦♥t❛r ❝❛rt❛s✿ ❍❛❜✐❧✐❞❛❞❡ ♦✉ ❈r✐♠❡❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

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✺ ❈r❛♣s ✹✷ ✺✳✶ ❆ ❤✐stór✐❛ ❞♦ ❈r❛♣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✷ ❆s r❡❣r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✷✳✶ ❚✐♣♦s ❞❡ ❆♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✺✳✸ ❆s ♣r♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✻ ❘♦❧❡t❛ ✹✼

✻✳✶ ❆ ❤✐stór✐❛ ❞♦ ❥♦❣♦ ❞❡ ❘♦❧❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✻✳✷ ❆♣♦st❛s ❡ Pr♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✷✳✶ ❆♣♦st❛s ✐♥t❡r♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻✳✷✳✷ ❆♣♦st❛s ❡①t❡r♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✻✳✷✳✸ ❱❛♥t❛❣❡♠ ❞❛ ❈❛s❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✼ ▼❡❣❛✲❙❡♥❛ ✺✸

✼✳✶ ❆ ❤✐stór✐❛ ❞❛ ▼❡❣❛✲❙❡♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✼✳✷ ❆s ❢♦r♠❛s ❞❡ ❛♣♦st❛r ❡ ❣❛♥❤❛r ♥❛ ▼❡❣❛✲❙❡♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✼✳✸ ❆s ♣r♦❜❛❜✐❧✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✼✳✸✳✶ ❙❡♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✼✳✸✳✷ ◗✉✐♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✼✳✸✳✸ ◗✉❛❞r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✽ ❈♦♥❝❧✉sã♦ ✺✼

✾ ❘❡❢❡rê♥❝✐❛s ✺✽

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■♥tr♦❞✉çã♦

❖s ❏♦❣♦s ❞❡ ❆③❛r s❡♠♣r❡ ✐♥tr✐❣❛r❛♠ ❛s ❝✐✈✐❧✐③❛çõ❡s✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ é♣♦❝❛ ♦✉ ❞❛ ❧♦❝❛❧✐③❛çã♦ ❣❡♦❣rá✜❝❛ ❞❡ t❛❧ ❝✐✈✐❧✐③❛çã♦✳ ❯♠❛ ✈❡③ q✉❡ ♦s ♣r✐♠❡✐r♦s ❥♦❣♦s s✉r❣✐r❛♠ ❞❡s❞❡ ❛ é♣♦❝❛ ❞♦s ■♠♣ér✐♦s ❘♦♠❛♥♦s ❡ ❝♦♥t✐♥✉❛♠ ❝❤❛♠❛♥❞♦ ❛ ❛t❡♥çã♦ ❞❡ ♣❡ss♦❛s ❛té ❤♦❥❡ ♣♦r ❧❛③❡r✱ ❡s♣♦rt❡ ♦✉ ❛té ❢♦♥t❡ ❞❡ r❡♥❞❛✱ ♦s ❏♦❣♦s ❞❡ ❆③❛r sã♦ ✐♥tr✐❣❛♥t❡s ♣♦r ❞✐✈❡rs♦s ❛s♣❡❝t♦s✳

❆t✉❛❧♠❡♥t❡ ❡①✐st❡♠ ✈ár✐♦s t✐♣♦s ❞❡ ❏♦❣♦s ❞❡ ❆③❛r✱ ♠❛s ♥♦ ❇r❛s✐❧ ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ é ♣r♦✐❜✐❞❛ ❞❡s❞❡ ✶✾✹✻✱ ❞✉r❛♥t❡ ♦ ●♦✈❡r♥♦ ❉✉tr❛✱ ♣r♦✐❜✐çã♦ ❡ss❛ r❡❧❛t❛❞❛ ♥♦ ❧✐✈r♦ ✧❆ ◆♦✐t❡ ❞♦ ▼❡✉ ❇❡♠✧✭❘✉② ●❛s♣❛r✮✱ ✜❝❛♥❞♦ s♦♠❡♥t❡ ❧✐❜❡r❛❞❛ ❛ ❡①♣❧♦r❛çã♦ ❞❡ss❡ t✐♣♦ ❞❡ ❥♦❣♦s ❛tr❛✈és ❞❛ ▲♦t❡r✐❛ ❋❡❞❡r❛❧✳ ❖s ❞❡♠❛✐s ❏♦❣♦s ❞❡ ❆③❛r sã♦ ❢❛❝✐❧♠❡♥t❡ ❡♥❝♦♥tr❛❞♦s ❡♠ ❝❛ss✐♥♦s✱ ❡s♣❛❧❤❛❞♦s ♣❡❧❛s ♠❛✐s ❞✐✈❡rs❛s ❧♦❝❛❧✐❞❛❞❡s ❞❡ ♠✉✐t♦s ♣❛ís❡s ❡♠ t♦❞♦s ♦s ❝♦♥t✐♥❡♥t❡s✳

❆ ❧❡❣❛❧✐③❛çã♦ ❞♦s ❏♦❣♦s ❞❡ ❆③❛r ✈❡♠ ❧❡✈❛♥t❛♥❞♦ ✈ár✐❛s ❞✐s❝✉ssõ❡s ♣♦❧ít✐❝❛s ❡ s♦❝✐❛✐s ♥♦ ❇r❛s✐❧✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❞✐♥❤❡✐r♦ q✉❡ ❞❡✐①❛ ❞❡ s❡r ❛rr❡✲ ❝❛❞❛❞❛ ❝♦♠ ✐♠♣♦st♦s ❡ ❛ ❝❧❛♥❞❡st✐♥✐❞❛❞❡ ❞♦s ❏♦❣♦s ❞❡ ❆③❛r s❡r r❡❧❛❝✐♦♥❛❞❛ ❛ ❝r✐♠❡s ❝♦♠♦ ❢❛❧s✐❞❛❞❡ ✐❞❡♦❧ó❣✐❝❛✱ ❧❛✈❛❣❡♠ ❞❡ ❞✐♥❤❡✐r♦✱ trá✜❝♦ ❞❡ ✐♥✢✉ê♥❝✐❛✱ trá✜❝♦ ❞❡ ❞r♦✲ ❣❛s ❡ ♣r♦st✐t✉✐çã♦✳ ❯♠❛ ✈❡③ q✉❡ ♦s ❥♦❣♦s ❝♦♠ ❛♣♦st❛s ♣♦❞❡♠ ♠♦✈✐♠❡♥t❛r ♠✉✐t♦ ❞✐♥❤❡✐r♦✱ ✈ár✐♦s ❥♦❣❛❞♦r❡s ❞❡♣❡♥❞❡♠ ❞❡❧❡s ❝♦♠♦ ♠❡✐♦ ❞❡ s✉st❡♥t♦✳ ❊♠ ❝♦♥tr❛♣❛r✲ t✐❞❛✱ ❥♦❣❛❞♦r❡s q✉❡ s❡ t♦r♥❛♠ ✈✐❝✐❛❞♦s✱ ❛❞q✉✐r✐♥❞♦ ✉♠❛ ♣❛t♦❧♦❣✐❛✱ ❝♦❧♦❝❛♠ ❡♠ r✐s❝♦ s✉❛ s❛ú❞❡ ❡ ♣❛tr✐♠ô♥✐♦ ♣ró♣r✐♦ ❡ ❞❡ ♣❡ss♦❛s ♣ró①✐♠❛s✳

❆ ❡s❝♦❧❤❛ ❞❡ss❡ t❡♠❛ ♣❛rt✐✉ ❞❡ ✉♠❛ ❝♦♥✈❡rs❛ ❝♦♠ ♦ ♦r✐❡♥t❛❞♦r ♣r♦❢✳ ❉r✳ ❆❧❡✲ ①❛♥❞r❡ ❞❡ ❇✉st❛♠❛♥t❡ ❙✐♠❛s✱ q✉❡ ♥❛ ♦♣♦rt✉♥✐❞❛❞❡ ❢❛❧á✈❛♠♦s ❞♦ ❢❛t♦ ❞❛ ♣♦♣✉❧❛çã♦ s❡r ❡♥❣❛♥❛❞❛ ❝♦♥st❛♥t❡♠❡♥t❡ ❡♠ ✈ár✐♦s ❛s♣❡❝t♦s✱ ✐♥❝❧✉s✐✈❡ ❡❞✉❝❛❝✐♦♥❛✐s✳ ❊ ❛❧❣✉♠❛s ♣❡ss♦❛s✱ ♠❡s♠♦ s❛❜❡♥❞♦ ❞❡ ❝❡rt♦s r✐s❝♦s ❡ ❞❡ ♣♦✉❝❛s ❝❤❛♥❝❡s ❞❡ ✈✐tór✐❛✱ ♣r❡❢❡r✐❛♠ ❛rr✐s❝❛r ❛té ♠❡s♠♦ ♦ q✉❡ ♥ã♦ t✐♥❤❛♠✳ ❋❛t♦ ❡ss❡ q✉❡ ♥♦s ❧❡✈♦✉ ❛ r❡✢❡t✐r ♦ ❝♦♠♣♦r✲ t❛♠❡♥t♦ ❞♦s ❥♦❣❛❞♦r❡s ❞❡ ❝❡rt♦s t✐♣♦s ❞❡ ❥♦❣♦s✱ ✐♥❝❧✉s✐✈❡ ♦s ❏♦❣♦s ❞❡ ❆③❛r✳

▼❛s✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❥♦❣♦s sã♦ ✐♠❡♥s♦s✳ ❊♥tã♦✱ ❞❡❝✐❞✐ ❢♦❝❛r ❡ss❡ tr❛❜❛❧❤♦ ❡♠ ❛❧❣✉♥s ❥♦❣♦s ❤✐st♦r✐❝❛♠❡♥t❡ ♠✉✐t♦ ♣♦♣✉❧❛r❡s ♥♦ ♠✉♥❞♦ ❡ ♦ ♠❛✐s ♣♦♣✉❧❛r ❞❡❧❡s ✭❧❡✲ ❣❛❧✐③❛❞♦✮ ♥♦ ❇r❛s✐❧✳

❊ss❡ tr❛❜❛❧❤♦ ✈❡rs❛rá✱ ♣♦rt❛♥t♦✱ s♦❜r❡ ♦s s❡❣✉✐♥t❡s ❏♦❣♦s ❞❡ ❆③❛r✿ ❥♦❣♦s ❞❡ ❝❛rt❛s ✭Pôq✉❡r ❡ ❇❧❛❝❦❥❛❝❦✮✱ ❥♦❣♦ ❞❡ ❞❛❞♦s ✭❈r❛♣s✮✱ ❥♦❣♦ ❞❡ ❘♦❧❡t❛ ❡ ❥♦❣♦ ❞❡ ❧♦t❡r✐❛ ✭▼❡❣❛✲ ❙❡♥❛✮✳ P❛r❛ ❝❛❞❛ ✉♠ ❞❡ss❡s ❥♦❣♦s✱ s❡rá ❝♦♥t❛❞❛ ✉♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❡ ❛ ❢♦r♠❛ ❝♦♠♦ s❡ ❞❡✈❡ ❥♦❣❛r✱ ❜❡♠ ❝♦♠♦ ❛s r❡❣r❛s✳

▲♦❣♦ ❛♣ós ❡①♣❧✐❝❛❞♦s ♦s ♣r♦❝❡❞✐♠❡♥t♦s ♣❛r❛ ❥♦❣❛r t❛✐s ❏♦❣♦s ❞❡ ❆③❛r✱ s❡rã♦ ♠♦str❛❞❛s ❛s ❢♦r♠❛s ❞❡ ❛♣♦st❛s ❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ s❡ ♦❜t❡r s✉❝❡ss♦ ✭♦✉ ✈✐tór✐❛✮ ♣❛r❛ ❝❛❞❛ t✐♣♦ ❞❡ ❛♣♦st❛ ❢❡✐t❛ ♣❡❧♦ ❥♦❣❛❞♦r✳

❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ tr❛❜❛❧❤♦ ♥ã♦ é ❡♥s✐♥❛r ❛ ❥♦❣❛r✱ ♠❛s s✐♠ ❛♥❛❧✐s❛r ❡ ♠♦str❛r q✉❡ ♦s ❏♦❣♦s ❞❡ ❆③❛r ❢♦r❛♠ ❝r✐❛❞♦s ♣❛r❛ ❢❛③❡r ❝♦♠ q✉❡ ♦s ❥♦❣❛❞♦r❡s ♣❡r❝❛♠✱ ✐♥❞❡♣❡♥❞❡♥t❡

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❞❡ s✉❛s ❤❛❜✐❧✐❞❛❞❡s ♦✉ ❡①♣❡r✐ê♥❝✐❛s ♥♦ ❥♦❣♦✱ ♠❛s ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❡ ❡st❛tíst✐❝❛s ❜❛s❡❛❞❛s ♥❛ ❧ó❣✐❝❛ ❞❛ t❡♦r✐❛ ❞♦s ❥♦❣♦s✳

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❈❛♣ít✉❧♦ ✶

❘❡✈✐s❛♥❞♦ ❝♦♥❝❡✐t♦s

❊st❡ ❝❛♣ít✉❧♦ tr❛t❛rá ❞❡ r❡✈✐s❛r ♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ❞❛ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛✲ tór✐❛ ❡ ❞❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ❜❡♠ ❝♦♠♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♣♦ss❛♠ s❡r út❡✐s ❛♦ ❧♦♥❣♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

✶✳✶ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛

❆ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛tór✐❛ é ♦ r❛♠♦ ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ ♣♦ss✐❜✐❧✐t❛✱ ❛tr❛✈és ❞❡ ❞❡t❡r✲ ♠✐♥❛❞❛s ♦♣❡r❛çõ❡s✱ ❛ ❝♦♥t❛❣❡♠ ❞❡ ❡❧❡♠❡♥t♦s ♣❛r❛ ❢♦r♠❛çã♦ ❞❡ ❝♦♥❥✉♥t♦s ❞✐st✐♥t♦s✱ s♦❜ ❛s ♠❛✐s ❞✐✈❡rs❛s ❝✐r❝✉♥stâ♥❝✐❛s✳

P♦❞❡♠♦s ❞❡st❛❝❛r✱ ♥❛ ❛♥á❧✐s❡ ❝♦♠❜✐♥❛tór✐❛✿

• Pr✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❝♦♥t❛❣❡♠

• ❋❛t♦r✐❛❧

• P❡r♠✉t❛çã♦ s✐♠♣❧❡s

• P❡r♠✉t❛çã♦ ❝♦♠ r❡♣❡t✐çã♦

• ❆rr❛♥❥♦ s✐♠♣❧❡s

• ❈♦♠❜✐♥❛çã♦ s✐♠♣❧❡s

• ❈♦♠❜✐♥❛çã♦ ❝♦♠ r❡♣❡t✐çã♦

✶✳✶✳✶ Pr✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❝♦♥t❛❣❡♠

Pr✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦

◗✉❛♥❞♦ ✉♠ ❡✈❡♥t♦ ♦❝♦rr❡ ❞❡ ♥ ♠❛♥❡✐r❛s s✉❝❡ss✐✈❛s ❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♦♥❞❡ ♥❛

i✲és✐♠❛ ❡t❛♣❛ ♦ ❡✈❡♥t♦ ♣♦❞❡ ♦❝♦rr❡r ❞❡ mi ♠♦❞♦s ❞✐❢❡r❡♥t❡s✱ t❡♠♦s q✉❡ ♦ t♦t❛❧ ❞❡

♠❛♥❡✐r❛s ❞♦ ❡✈❡♥t♦ ♦❝♦rr❡r é ♦ ♣r♦❞✉t♦ t♦❞❛s ❛s ❡t❛♣❛s✱ ❞❛ ♣r✐♠❡✐r❛ ❛té ❛ ♥✲és✐♠❛✿

m1·m2·m3· · ·mn✳

❊①❡♠♣❧♦✳ ◗✉❛♥t♦s ♥ú♠❡r♦s í♠♣❛r❡s ❞❡ três ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s ❡①✐st❡♠✱ s❡♥❞♦ q✉❡ ❧✐❞♦s ❞❡ trás ♣❛r❛ ❢r❡♥t❡ ❢♦r♠❛♠ ♥ú♠❡r♦s ♣❛r❡s❄

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✶✳✶✳ ❆◆➪▲■❙❊ ❈❖▼❇■◆❆❚Ó❘■❆

❙♦❧✉çã♦✳ ❙❡ ♦ ♥ú♠❡r♦ ❝♦♥té♠ ✸ ❛❧❣❛r✐s♠♦s✱ ✈❛♠♦s ❝❤❛♠❛r ♦ ❛❧❣❛r✐s♠♦ ❞❛ ❝❡♥✲ t❡♥❛ ❞❡ m1✱ ♦ ❛❧❣❛r✐s♠♦ ❞❛ ❞❡③❡♥❛ ❞❡ m2 ❡ ♦ ❛❧❣❛r✐s♠♦ ❞❛ ✉♥✐❞❛❞❡ ❞❡ m3✳ ❙❡ ♦

♥ú♠❡r♦ é í♠♣❛r✱ ♦ ❛❧❣❛r✐s♠♦ ❞❛ ✉♥✐❞❛❞❡ só ♣♦❞❡ s❡r ❝♦♠♣♦st♦ ♣♦r ✉♠ ❞♦s ❛❧❣❛r✐s✲ ♠♦s✿ ✶✱ ✸✱ ✺✱ ✼ ♦✉ ✾✳ ▲♦❣♦✱m3 = 5✳ ❈♦♠♦ ♦ ♥ú♠❡r♦ ❡♠ q✉❡stã♦✱ q✉❛♥❞♦ ❧✐❞♦ ❞❡ trás

♣❛r❛ ❢r❡♥t❡ ❢♦r♠❛ ✉♠ ♥ú♠❡r♦ ♣❛r✱ ♦ ❛❧❣❛r✐s♠♦ ❞❛ ❝❡♥t❡♥❛ ❞❡✈❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ❛❧❣❛r✐s♠♦ ♣❛r ❡s❝♦❧❤✐❞♦ ❡♥tr❡✿ ✷✱ ✹✱ ✻ ♦✉ ✽ ✭♣♦✐s ♥❡♥❤✉♠ ♥ú♠❡r♦ ❝♦♠❡ç❛ ❝♦♠ ♦ ❛❧❣❛r✐s♠♦ ✵✮✳ ▲♦❣♦✱ m1 = 4✳ ❖ ❛❧❣❛r✐s♠♦ ❞❛ ❞❡③❡♥❛ ❞❡✈❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠

❛❧❣❛r✐s♠♦ ❞♦ s✐st❡♠❛ ❞❡❝✐♠❛❧ ❡✱ ❛✐♥❞❛✱ ❞❡✈❡ s❡r ❞✐❢❡r❡♥t❡ ❞♦ ❛❧❣❛r✐s♠♦ ❡s❝♦❧❤✐❞♦ ♣❛r❛ ❛s ♣♦s✐çõ❡sm3 ❡m1✳ ▲♦❣♦✱ m2 = 8✳ P♦rt❛♥t♦✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♥ú♠❡r♦s í♠♣❛r❡s ❞❡

✸ ❛❧❣❛r✐s♠♦s✱ q✉❡ q✉❛♥❞♦ ❧✐❞♦s ❞❡ trás ♣❛r❛ ❢r❡♥t❡ ❢♦r♠❛♠ ✉♠ ♥ú♠❡r♦ ♣❛r é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ m1.m2.m3 = 4.8.5 = 160♥ú♠❡r♦s✳

❊①❡♠♣❧♦✳ ◗✉❛♥t♦s ❞✐✈✐s♦r❡s ♥❛t✉r❛✐s t❡♠ ♦ ♥ú♠❡r♦ ✺✹✵✵❄ ◗✉❛♥t♦s ❞❡❧❡s sã♦ í♠♣❛r❡s❄ ❆❧❣✉♠ ❞❡❧❡s é q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦❄

❙♦❧✉çã♦✳ P❛r❛ r❡s♦❧✈❡r ❡ss❡ ♣r♦❜❧❡♠❛✱ ♣r❡❝✐s❛♠♦s ❢❛t♦r❛r ♦ ✺✹✵✵✱ ❡ ❧♦❣♦ ❡♥❝♦♥✲ tr❛♠♦s q✉❡ 5400 = 23_.₃3_.₅2_{✱ ❡ t♦❞♦s ♦s ❞✐✈✐s♦r❡s ❞❡ ✺✹✵✵ sã♦✱ ♣♦rt❛♥t♦✱ ❞❛ ❢♦r♠❛} 2x_.₃y_.₅z_{✱ ♦♥❞❡} _x

∈0,1,2,3, y _∈0,1,2,3 e z _∈0,1,2✳

∗ ▲♦❣♦✱ ❤á ✹✳✹✳✸ ❂ ✹✽ ❡s❝♦❧❤❛s ❞✐❢❡r❡♥t❡s ❞❡ ❡①♣♦❡♥t❡s ✱ ♣♦rt❛♥t♦ ♦ ♥ú♠❡r♦ ❞❡

❞✐✈✐s♦r❡s ❞❡ ✺✹✵✵ é ✹✽✳

∗P❛r❛ ♦ ❞✐✈✐s♦r s❡r í♠♣❛r✱ ❡❧❡ ❞❡✈❡ s❡r ❞❛ ❢♦r♠❛ 20_.₃y_.₅z_{✳ ❚❡♠♦s ❡♥tã♦✿ ✶✳✹✳✸ ❂}

✶✷ ❞✐✈✐s♦r❡s ❞❡ ✺✹✵✵ q✉❡ sã♦ í♠♣❛r❡s✳

∗P❛r❛ ♦ ❞✐✈✐s♦r s❡r q✉❛❞r❛❞♦ ♣❡r❢❡✐t♦✱ ①✱② ❡ ③ tê♠ q✉❡ s❡r ♥ú♠❡r♦s ♣❛r❡s✳ ❊♥tã♦✱

✷✳✷✳✷ ❂ ✽ ❞✐✈✐s♦r❡s ❞❡ ✺✹✵✵ sã♦ q✉❛❞r❛❞♦s ♣❡r❢❡✐t♦s✳

Pr✐♥❝í♣✐♦ ❛❞✐t✐✈♦

❙✉♣♦♥❤❛ q✉❡ ✉♠ ❡✈❡♥t♦ ♣♦ss❛ ♦❝♦rr❡r ❡♠ k s✐t✉❛çõ❡s✳ ❙✉♣♦♥❤❛✱ ❛✐♥❞❛✱ ✉❡ ♥❛ i✲és✐♠❛ s✐t✉❛çã♦✱ ♦ ♦ ❡✈❡♥t♦ ♣♦❞❡ ♦❝♦rr❡r ❞❡ ni ❢♦r♠❛s✳ ❋✐♥❛❧♠❡♥t❡✱ s✉♣♦♥❤❛ ✉❡

❡✈❡♥t♦s ❞❡ s✐t✉❛çõ❡s ❞✐st✐♥t❛s ♥ã♦ ♣♦❞❡♠ ♦❝♦rr❡r s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❊♥tã♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❢♦r♠❛s ❞♦ ❡✈❡♥t♦ ♦❝♦rr❡r ❡♠ ❛❧❣✉♠❛ ❞❛s s✐t✉❛çõ❡s én1+n2+· · ·+nk✳

❊①❡♠♣❧♦✳ ❯♠❛ ❧❛♥❝❤♦♥❡t❡ ♦❢❡r❡❝❡✱ ❡♠ s❡✉ ❝❛r❞á♣✐♦✱ ✸ ♦♣çõ❡s ❞❡ s❛♥❞✉í❝❤❡s ✭❝❛r♥❡✱ ❢r❛♥❣♦ ♦✉ ❤♦t ❞♦❣✮✱ ✸ ♦♣çõ❡s ❞❡ ❜❡❜✐❞❛s ✭á❣✉❛✱ s✉❝♦ ♦✉ r❡❢r✐❣❡r❛♥t❡✮ ❡ ✷ ♦♣✲ çõ❡s ❞❡ s♦❜r❡♠❡s❛ ✭s♦r✈❡t❡ ♦✉ ♣❛✈ê✮✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ✉♠ ❝❧✐❡♥t❡ ♣♦❞❡ ❝♦♥s✉♠✐r ✉♠ ❧❛❝❤❡ ❝♦♠♣❧❡t♦ ✭s❛♥❞✉í❝❤❡✱ ❜❡❜✐❞❛ ❡ s♦❜r❡♠❡s❛✮✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❡❧❡ ❝♦♠❛ ♣❛✈ê ♦✉ ♥ã♦ t♦♠❡ r❡❢r✐❣❡r❛♥t❡❄

❙♦❧✉çã♦✳ P❛r❛ ❡ss❛ s✐t✉❛çã♦✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❝❛❞❛ ♦♣çã♦ s❡♣❛r❛❞❛♠❡♥t❡✿ ♥❛ ♣r✐♠❡✐r❛✱ ♦ ❝❧✐❡♥t❡ ❝♦♠❡ ♣❛✈ê ❡✱ ♥❛ s❡❣✉♥❞❛✱ ♦ ❝❧✐❡♥t❡ ♥ã♦ t♦♠❛ r❡❢r✐❣❡r❛♥t❡✳

∗ ❆s ♠❛♥❡✐r❛s ❞♦ ❝❧✐❡♥t❡ ♣❡❞✐r ✉♠ ❧❛♥❝❤❡ ❝♦♠♣❧❡t♦✱ ❝♦♠❡♥❞♦ ♣❛✈ê✱ sã♦ ❞❡ ✸✳✸✳✶

❂ ✾ ♠♦❞♦s ❞✐❢❡r❡♥t❡s✳

∗❆s ♠❛♥❡✐r❛s ❞♦ ❝❧✐❡♥t❡ ♣❡❞✐r ✉♠ ❧❛♥❝❤❡ ❝♦♠♣❧❡t♦✱ s❡♠ t♦♠❛r r❡❢r✐❣❡r❛♥t❡✱ sã♦

❞❡ ✸✳✷✳✷ ❂ ✶✷ ♠♦❞♦s ❞✐❢❡r❡♥t❡s✳

▲♦❣♦✱ ❤á ✾ ✰ ✶✷ ❂ ✷✶ ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❞♦ ❝❧✐❡♥t❡ ♣❡❞✐r ✉♠ ❧❛♥❝❤❡ ❝♦♠❡♥❞♦ ♣❛✈ê ♦✉ s❡♠ t♦♠❛r r❡❢r✐❣❡r❛♥t❡✳

(14)

✶✳✶✳ ❆◆➪▲■❙❊ ❈❖▼❇■◆❆❚Ó❘■❆

✶✳✶✳✷ ❋❛t♦r✐❛❧

❘❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ ✦✱ ♦ ❢❛t♦r✐❛❧ ❞❡ ✉♠ ♥ú♠❡r♦ ♥ é ♦ ♣r♦❞✉t♦ ❞❡ t♦❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❞❡ ✶ ❛té ♥✳ ❖✉ ❛✐♥❞❛✿

n! = n

Y

k=1

k, _∀n_∈N

❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ss❛ ❞❡✜♥✐çã♦✱ t❡♠♦s q✉❡✿

• ✶✦ ❂ ✶

• ✭♥✰✶✮✦ ❂ ✭♥✰✶✮✳♥✦

• P♦r ❝♦♥✈❡♥çã♦✱ ✵✦ ❂ ✶

➱ ❝♦♥✈❡♥✐❡♥t❡ s❛❧✐❡♥t❛r q✉❡ ♦s ❢❛t♦r✐❛✐s ♣♦❞❡♠ s❡r ❡st❡♥❞✐❞♦s ♣❛r❛ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥ã♦✲♥❛t✉r❛✐s✱ ❞❡s❞❡ q✉❡ ♥ã♦ s❡❥❛♠ ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✳ P❛r❛ z _∈ C_\

{−1,₋2, . . ._}✱ ❝♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ ❣❛♠❛✿

Γ(z+ 1) =

Z ∞

0

tz_e−_t

dt

♦♥❞❡✱ s❡n é ♥❛t✉r❛❧✱ Γ(n+ 1) =n!✳

❏á ♣❛r❛ ❢❛t♦r✐❛✐s ❞❡ ♥ú♠❡r♦s ❣r❛♥❞❡s✱ ❞✐❣❛♠♦sn✱ ♣♦❞❡✲s❡ ✉s❛r ❛ ❛♣r♦①✐♠❛çã♦ ❞❡

❙t✐r❧✐♥❣✿

n!_∼√2πnn e

n

➱ ✐♥t❡r❡ss❛♥t❡ ♦❜s❡r✈❛r q✉❡✱ ♣❛r❛ ♦s ❢❛t♦r✐❛✐s✱ ♥ã♦ sã♦ ✈á❧✐❞❛s ❛s ♦♣❡r❛çõ❡s ❛r✐t✲ ♠ét✐❝❛s ❞❡ ❛❞✐çã♦✿ ♥✦ ✰ ♠✦ 6= ✭♥✰♠✮✦✱ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✿ ♥✦✳♠✦ ₆= ✭♥✳♠✮✦✳ ❆♣❡♥❛s

❝♦♥s❡❣✉✐♠♦s s✐♠♣❧✐✜❝❛r ♦s ❢❛t♦r✐❛✐s✱ s❡ ❛t❡♥t❛r♠♦s ♣❛r❛ ❛s s✉❛s ❡①♣❛♥sõ❡s✿ ❊①❡♠♣❧♦✳ ❙✐♠♣❧✐✜q✉❡ ❛ ❢r❛çã♦ 9!

6!✳

❙♦❧✉çã♦✳ ❆♦ ❝♦♥trár✐♦ ❞❛ s✐♠♣❧✐✜❝❛çã♦ ❛r✐t♠ét✐❝❛✱ ❡ss❛ s✐♠♣❧✐✜❝❛çã♦ ♥ã♦ é ✐❣✉❛❧ ❛(3₂)!✳ ❆♦ ❡①♣❛♥❞✐r ♦ ✾✦ ❡ ♦ ✻✦✱ t❡♠♦s✿ 9!_6! = 1.2.3.4.5.6.7.8.9

1.2.3.4.5.6 = 7.8.9 = 504✳

❊①❡♠♣❧♦✳ ❘❡s♦❧✈❛ ❛ ❡q✉❛çã♦ (n+3)! (n+1)! = 56✳

❙♦❧✉çã♦✳ P❛r❛ r❡s♦❧✈❡r ❡ss❡ t✐♣♦ ❞❡ ❡q✉❛çã♦✱ ❞❡✈❡♠♦s ♦❜s❡r✈❛r ♦ ♠❛✐♦r ❞♦s ❢❛t♦r✐❛✐s ✭♥✰✸✮✦✱ ❜❡♠ ❝♦♠♦ s✉❛ ❡①♣❡♥sã♦✿ ✭♥✰✸✮✦ ❂ ✭♥✰✸✮✳✭♥✰✷✮✳✭♥✰✶✮✳♥✳· · ·✳✸✳✷✳✶✱

♦✉ ❛✐♥❞❛✿ ✭♥✰✸✮✦ ❂ ✭♥✰✸✮✳✭♥✰✷✮✳✭♥✰✶✮✦✳ ❊♥tã♦✱

(n+3)! (n+1)! =

(n+3).(n+2).(n+1)!

(n+1)! = 56 ⇒

⇒(n+ 3).(n+ 2) = 56_⇒n2_{+ 5}_n_{+ 6 = 56}_⇒_n2_{+ 5}_n₋_{50 = 0}

▲♦❣♦✱ ♥ ❂ ✲✶✵ ♦✉ ♥ ❂ ✺✳ ▼❛sn _∈N✱ ❡♥tã♦ ❛ ú♥✐❝❛ s♦❧✉çã♦ ♣❛r❛ ❡ss❛ ❡q✉❛çã♦ é ♥❂✺✳

❍á ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ♦s ❢❛t♦r✐❛✐s✱ ❞❡♥tr❡ ❡❧❛s ❛s ♣❡r♠✉t❛çõ❡s✱ ♦s ❛rr❛♥❥♦s ❡ ❛s ❝♦♠❜✐♥❛çõ❡s✳

(15)

✶✳✶✳ ❆◆➪▲■❙❊ ❈❖▼❇■◆❆❚Ó❘■❆

✶✳✶✳✸ P❡r♠✉t❛çõ❡s

P❡r♠✉t❛çã♦ s✐♠♣❧❡s

❆ ♣❛❧❛✈r❛ P❡r♠✉t❛çã♦ s✐❣♥✐✜❝❛ tr♦❝❛ ❞❡ ♣♦s✐çõ❡s✳ P❛r❛ ♣❡r♠✉t❛r ♥ ❡❧❡♠❡♥t♦s✱ ❡♠ q✉❡ ♥ã♦ ❤á r❡♣❡t✐çã♦ ❞❡ ♥❡♥❤✉♠ ❞❡❧❡s ✭❞❛í ❛ ❡①♣r❡ssã♦ ✧P❡r♠✉t❛çã♦ s✐♠♣❧❡s✧✮✱ ❞❡✈❡♠♦s ✉t✐❧✐③❛r✿

Pn =n!

❯♠❛ ✈❡③ q✉❡ ❞✐s♣♦♠♦s ❞❡ ♥ ❡❧❡♠❡♥t♦s ♣❛r❛ ♦❝✉♣❛r ♥ ♣♦s✐çõ❡s✱ ♣❛r❛ ❛ ♣r✐♠❡✐r❛ ♣♦s✐çã♦ x1 t❡♠♦s ♥ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡s❝♦❧❤❛✱ ♣❛r❛ ❛ s❡❣✉♥❞❛ ♣♦s✐çã♦ x2 t❡♠♦s ♥✲✶

♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡s❝♦❧❤❛✱ ♣❛r❛ ❛ t❡r❝❡✐r❛ ♣♦s✐çã♦x3 t❡♠♦s ♥✲✷ ❡❧❡♠❡♥t♦s ♣♦ssí✈❡✐s ❞❡

❡s❝♦❧❤❛✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❛té ❛ ú❧t✐♠❛ ♣♦s✐çã♦xn✱ ❛ q✉❛❧ só t❡rá r❡st❛❞♦ ✉♠

ú♥✐❝♦ ❡❧❡♠❡♥t♦ ♣♦ssí✈❡❧ ❞❡ ❡s❝♦❧❤❛ ♣❛r❛ ❡ss❛ ♣♦s✐çã♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱

Pn=n.(n−1).(n−2).· · · .2.1 =n!

❊①❡♠♣❧♦✳ ❯♠❛ ❢❛♠í❧✐❛ ✭♣❛✐✱ ♠ã❡ ❡ três ✜❧❤♦s✮ q✉❡r❡♠ t✐r❛r ✉♠❛ ❢♦t♦ ♣❛r❛ ♦ á❧❜✉♠ ❞❡ r❡❝♦r❞❛çõ❡s✳ P❛r❛ ✐ss♦✱ ♦ ❢♦tó❣r❛❢♦ s✉❣❡r✐✉ q✉❡ ♦ ♣❛✐ ✜❝❛ss❡ ❛♦ ❧❛❞♦ ❞❛ ♠ã❡ s❡♥t❛❞♦s ❡ ♦s três ✜❧❤♦s ✜❝❛ss❡♠ ❥✉♥t♦s ❡♠ ♣é✱ ♦✉ ♦s ♣❛✐s ❡♠ ♣é ❡ ♦s ✜❧❤♦s s❡♥t❛❞♦s✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ❡ss❛ ❢♦t♦ ♣♦❞❡ s❡r t✐r❛❞❛❄

❙♦❧✉çã♦✳ ❖ ♣❛✐ ❡ ❛ ♠ã❡ ❞❡✈❡♠ ✜❝❛r ❥✉♥t♦s✱ ❡♥tã♦ ❡❧❡s ♣♦❞❡♠ ❧✐✈r❡♠❡♥t❡ tr♦❝❛r ❞❡ ❧✉❣❛r✱ ❧♦❣♦ ✷✦ ❂ ✷✳ ❏á ♦s três ✜❧❤♦s ❞❡✈❡♠ ✜❝❛r ❥✉♥t♦s✱ ♠❛s t❛♠❜é♠ ♣♦❞❡♠ tr♦❝❛r ❞❡ ❧✉❣❛r✱ ❧♦❣♦ ✸✦ ❂ ✻✳ ❊✱ t❛♠❜é♠✱ ❤á ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞♦s q✉❡ ❡stã♦ s❡♥t❛❞♦s tr♦❝❛r❡♠ ❞❡ ❧✉❣❛r ❝♦♠ ♦s q✉❡ ❡stã♦ ❡♠ ♣é✱ ♦✉ s❡❥❛✱ ✷✦ ❂ ✷✳ P♦rt❛♥t♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ❤á ✷✳✻✳✷ ❂ ✷✹ ♠❛♥❡✐r❛s ❞✐❢❡r❡♥t❡s ❞❛ ❢❛♠í❧✐❛ t✐r❛r ❡ss❛ ❢♦t♦✳

❊①❡♠♣❧♦✳ ◗✉❛♥t♦s ❛♥❛❣r❛♠❛s t❡♠ ❛ ♣❛❧❛✈r❛ ❇❊❘▼❯❉❆❄ ◗✉❛♥t♦s ❞❡ss❡s ❛♥❛✲ ❣r❛♠❛s ❝♦♠❡ç❛♠ ♣♦r ✈♦❣❛❧ ❡ t❡r♠✐♥❛♠ ❝♦♠ ❝♦♥s♦❛♥t❡❄

❙♦❧✉çã♦✳ ❆♥❛❣r❛♠❛s sã♦ ♥♦✈❛s ❡str✉t✉r❛s ❞❡ ✉♠❛ ♣❛❧❛✈r❛ ✭♥ã♦ ♣r❡❝✐s❛♠ ❢❛③❡r s❡♥t✐❞♦ ❛ ♥í✈❡❧ ❞❡ ❧✐♥❣✉íst✐❝❛✮✱ ♠✉❞❛♥❞♦ ❛♣❡♥❛s ❞❡ ♣♦s✐çã♦ s✉❛s ❧❡tr❛s✳ ❉❡ss❛ ❢♦r♠❛✱ ❡ss❛ ♠✉❞❛♥ç❛ ❞❡ ♣♦s✐çã♦ ♥♦s r❡♠❡t❡ ❛ ✐❞❡✐❛ ❞❡ ♣❡r♠✉t❛çõ❡s✳ ❊♥tã♦✱ ♣❛r❛ s❛❜❡r ♦ t♦t❛❧ ❞❡ ❛♥❛❣r❛♠❛s ❞❛ ♣❛❧❛✈r❛ ❇❊❘▼❯❉❆✱ ♣r❡❝✐s❛♠♦s ♣❡r♠✉t❛r s✉❛s ✼ ❧❡tr❛s✱ ❧♦❣♦✿

P7 = 7! = 5040 ❛♥❛❣r❛♠❛s✳ ❏á ♣❛r❛ ♦ 2o q✉❡st✐♦♥❛♠❡♥t♦✱ ❞❡✈❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛

♣r✐♠❡✐r❛ ❧❡tr❛ ❞❡✈❡ s❡r ✉♠❛ ✈♦❣❛❧✱ ♦✉ s❡❥❛✱m1 ∈ {A, E, U}✱ ❛ ú❧t✐♠❛ ❧❡tr❛ ❞❡✈❡ s❡r

✉♠❛ ❝♦♥s♦❛♥t❡✱ ♦✉ s❡❥❛✱ m7 ∈ {B, D, M R}✱ ❡ ❛s ❧❡tr❛s ❞♦ ♠❡✐♦ ♣♦❞❡♠ ♣❡r♠✉t❛r

❧✐✈r❡♠❡♥t❡ ❞❡♥tr❡ ❛s ❧❡tr❛s q✉❡ ✜❝❛r❛♠ ✧❞✐s♣♦♥í✈❡✐s✧✳ ❊♥tã♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛✲ ♠❡♥t❛❧ ❞❛ ❝♦♥t❛❣❡♠✱ ♦ t♦t❛❧ ❞❡ ❛♥❛❣r❛♠❛s ❞❛ ♣❛❧❛✈r❛ ❜❡r♠✉❞❛ q✉❡ ❝♦♠❡ç❛♠ ❝♦♠ ✈♦❣❛❧ ❡ t❡r♠✐♥❛♠ ❝♦♠ ❝♦♥s♦❛♥t❡ é ✐❣✉❛❧ ❛✿ 3.P5.4 = 3.5!.4 = 1440 ❛♥❛❣r❛♠❛s✳

P❡r♠✉t❛çã♦ ❝♦♠ r❡♣❡t✐çã♦

❆❧❣✉♠❛s s✐t✉❛çõ❡s✱ ❡①✐❣❡♠ q✉❡ ♣❡♥s❡♠♦s ❛❧é♠ ❞❛s ♣❡r♠✉t❛çõ❡s s✐♠♣❧❡s✱ ♣♦✐s ❡①✐st❡♠ ♣❡q✉❡♥♦s ❡♥tr❛✈❡s✳ ➱ ♦ ❝❛s♦ ✭♣♦r ❡①❡♠♣❧♦✮ ❞❡ ♣❛❧❛✈r❛s q✉❡ ♣♦ss✉❡♠ ❧❡tr❛s r❡♣❡t✐❞❛s✱ ♣♦✐s ❛♦ tr♦❝❛r♠♦s ❡ss❛s ❧❡tr❛s r❡♣❡t✐❞❛s ❞❡ ♣♦s✐çã♦✱ ♥ã♦ ❢♦r♠❛r❡♠♦s ✉♠❛ ♥♦✈❛ ❡str✉t✉r❛ ❝♦♠ ❡ss❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ❧❡tr❛s✳

P❛r❛ ♦❜t❡r ♦ t♦t❛❧ ❞❡ ♣❡r♠✉t❛çõ❡s ❞❡ ♥ ♦❜❥❡t♦s✱ s❡♥❞♦ q✉❡ ❡st❡s ♣♦ss✉❡♠ r❡♣❡✲ t✐çõ❡s ❞❡α, β, γ,_{· · ·} ♦❜❥❡t♦s✱ ✉t✐❧✐③❛♠♦s✿

(16)

✶✳✶✳ ❆◆➪▲■❙❊ ❈❖▼❇■◆❆❚Ó❘■❆

Pα,β,γ,···

n =

n!

α!.β!.γ!._{· · ·}

P❛r❛ ♣r♦✈❛r♠♦s ❛ ❢ór♠✉❧❛ ❛❝✐♠❛✱ ♦❜s❡r✈❡ q✉❡ ❞❛❞❛ ✉♠❛ ♣❛❧❛✈r❛ ♦♥❞❡ ✉♠❛ ❝❡rt❛ ❧❡tr❛✱ ❞❡♥♦t❡✲❛ ♣♦r x✱ r❡♣❡t❡✲s❡ α ✈❡③❡s✱ t❡♠♦s α! ❢♦r♠❛s ❞❡ tr♦❝❛r ❛ ❧❡tr❛ x ❞❡

❧✉❣❛r s❡♠ ♠✉❞❛r ❛ ♣❛❧❛✈r❛ ✭tr♦❝❛♥❞♦✲❛s ❞❡ ♣♦s✐çã♦ ❡♥tr❡ s✐✮✳ ▲♦❣♦✱ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s n! ♣❡r♠✉t❛çõ❡s ❞❛ ♣❛❧❛✈r❛ ♦r✐❣✐♥❛❧✱ t❡♠♦s α! ♣❛❧❛✈r❛s r❡♣❡t✐❞❛s ✭❛♣❡♥❛s

❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ❛s ♠✉❞❛♥ç❛s ❢❡✐t❛s ❝♦♠ r❡❧❛çã♦ à ❧❡tr❛ x✮✳ ❘❡♣❡t✐♥❞♦✲

s❡ ♦ r❛❝✐♦❝í♥✐♦ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❧❡tr❛s q✉❡ s❡ r❡♣❡t❡♠✱ ❡ ❛♣❧✐❝❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ♦❜t❡♠♦s q✉❡ ❝❛❞❛ ♣❛❧❛✈r❛ ❞❛s n! ♣❡r♠✉t❛çõ❡s ❞❛ ♣❛❧❛✈r❛ ♦r✐❣✐♥❛❧✱

r❡♣❡t❡✲s❡α!β!γ!_{· · ·} ✈❡③❡s✳ P♦rt❛♥t♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❛❧❛✈r❛s ❞✐st✐♥t❛s é✿

n!

α!.β!.γ!._{· · ·}.

❊①❡♠♣❧♦✳ P♦ss✉♦ ✺ ❜♦❧❛s ✈❡r♠❡❧❤❛s✱ ✸ ❜♦❧❛s ❛③✉✐s ❡ ✷ ❜♦❧❛s ✈❡r❞❡s ♣❛r❛ ❛rr✉♠á✲ ❧❛s ❡♠ ✉♠ ♠✉r❛❧✱ ❢♦r♠❛♥❞♦ ❛♣❡♥❛s ✉♠❛ ❝♦❧✉♥❛ ❞❡ ❜♦❧❛s✳ ◗✉❛♥t♦s ♠✉r❛✐s ❞✐❢❡r❡♥t❡s ♣♦ss♦ ❢♦r♠❛r❄

❙♦❧✉çã♦✳ ◆❡ss❡ ❝❛s♦✱ q✉❡r❡♠♦s ♣❡r♠✉t❛r ✶✵ ❜♦❧❛s✱ ❞❛s q✉❛✐s ❛s ❝♦r❡s ❞❡ ✺✱ ✸ ❡ ✷ ❞❡❧❛s s❡ r❡♣❡t❡♠✳ ❆♣❧✐❝❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❛s ♣❡r♠✉t❛çõ❡s ❝♦♠ r❡♣❡t✐çõ❡s✿

P₁₀5,3,2 = 10! 5!.3!.2! =

10.9.8.7.6

3.2.1.2.1 = 2520 ♠✉r❛✐s ❞✐❢❡r❡♥t❡s✳

❊①❡♠♣❧♦✳ ❉♦✐s ❛♠✐❣♦s ❞❡❝✐❞✐r❛♠ ❥♦❣❛r ①❛❞r❡③ ❞❡ ✉♠❛ ❢♦r♠❛ ❞✐❢❡r❡♥t❡✿ ❝❛❞❛ ✉♠ ❞❡❧❡s✱ ❡♠ s✉❛s ❞✉❛s ♣r✐♠❡✐r❛s ❧✐♥❤❛s ✭✶✻ ❝❛s❛s✮ ♣♦❞❡♠ ❞✐s♣♦r ❞❛ ❛rr✉♠❛çã♦ q✉❡ q✉✐s❡r❡♠ ❞❡ s✉❛s ♣❡ç❛s✳ ◗✉❛♥t❛s ❛rr✉♠❛çõ❡s ❝❛❞❛ ✉♠ ❞♦s ❛♠✐❣♦s ♣♦❞❡ ❢❛③❡r ♣❛r❛ ❝♦♠❡ç❛r ♦ ❥♦❣♦❄

❙♦❧✉çã♦✳ ❈❛❞❛ ❛♠✐❣♦ ❞✐s♣õ❡ ❞❡ ✶✻ ♣❡ç❛s✱ s❡♥❞♦ ❡❧❛s ✽ ♣❡õ❡s✱ ✷ t♦rr❡s✱ ✷ ❝❛✈❛❧♦s✱ ✷ ❜✐s♣♦s✱ ✶ r❛✐♥❤❛ ❡ ✶ r❡✐✳ ❉❡ss❛ ❢♦r♠❛✱ ❛♦ tr♦❝❛r♠♦s ♣❡ç❛s ✐❣✉❛✐s✱ ♥ã♦ ❢♦r♠❛♠♦s ✉♠❛ ♥♦✈❛ ❛rr✉♠❛çã♦ ❞♦ t❛❜✉❧❡✐r♦✳ ❊♥tã♦✿

P₁₆8,2,2,2 = _8!_._2!16!_._2!_._2! = 64864800 ❛rr✉♠❛çõ❡s ❞✐st✐♥t❛s✳

✶✳✶✳✹ ❆❣r✉♣❛♠❡♥t♦s

❆♦ ❡s❝♦❧❤❡r ✈ár✐♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❣r✉♣♦✱ ❡st❛♠♦s ❛❣r✉♣❛♥❞♦ ❡ss❡s ❡❧❡♠❡♥t♦s✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦ ♠♦❞♦ ❝♦♠♦ ❢❛③❡♠♦s t❛✐s ❡s❝♦❧❤❛s ♣♦❞❡ ✐♥t❡r❢❡r✐r ♥♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦✳ ❉❡st❛ ❢♦r♠❛✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠ t♦t❛❧ ❞❡ ♥ ❡❧❡♠❡♥t♦s✱ ❞♦s q✉❛✐s ❡s❝♦✲ ❧❤❡r❡♠♦s ♣ ❡❧❡♠❡♥t♦s✳

❆rr❛♥❥♦ s✐♠♣❧❡s

◗✉❛♥❞♦ ❡s❝♦❧❤❡♠♦s ♣ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ t♦t❛❧ ❞❡ ♥ ❡❧❡♠❡♥t♦s ✭❝♦♠n _≥p✮✱ ❞❡ t❛❧

❢♦r♠❛ q✉❡ ❛ ♦r❞❡♠ ❞❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❡s❝♦❧❤✐❞♦ ✐♥✢✉❡♥❝✐❛ ♥♦ ❣r✉♣♦ ❢♦r♠❛❞♦✱ ❡st❛♠♦s ❞✐❛♥t❡ ❞❡ ✉♠ ❆rr❛♥❥♦ s✐♠♣❧❡s✳ P❛r❛ ❞❡t❡r♠✐♥❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❣r✉♣♦s✱ ❡♠ q✉❡ ❛ ♦r❞❡♠ ❢❛③ ❞✐❢❡r❡♥ç❛✱ ♣♦❞❡♠♦s ✉s❛r✿

An,p =

n! (n₋p)!

(17)

✶✳✶✳ ❆◆➪▲■❙❊ ❈❖▼❇■◆❆❚Ó❘■❆

❖s ❛rr❛♥❥♦s s✐♠♣❧❡s t❛♠❜é♠ ♣♦❞❡♠ s❡r ❡♥t❡♥❞✐❞♦s ❝♦♠♦ ♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛✲ t✐✈♦ ❞♦ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❝♦♥t❛❣❡♠✱ ♣♦✐s ❝❛❞❛ ❡❧❡♠❡♥t♦ t❡♠ s✉❛ ♣♦s✐çã♦ ✭❡ ú♥✐❝❛✱ ✉♠❛ ✈❡③ q✉❡ ❛ ✈❛r✐❛çã♦ ❞❡ ♣♦s✐çã♦ ♠♦❞✐✜❝❛ ♦ ❛❣r✉♣❛♠❡♥t♦✮✳

◗✉❛♥❞♦ t❡♠♦s ♥ ❡❧❡♠❡♥t♦s ✭t♦❞♦s ❞✐st✐♥t♦s ❡♥tr❡ s✐✮ ❡ q✉❡r❡♠♦s ❛❣r✉♣á✲❧♦s ❡♠ ♣ ❢♦r♠❛çõ❡s ❞✐st✐♥t❛s✱ ♣❛r❛ ♦ ♣r✐♠❡✐r❛ ♣♦s✐çã♦ ✭x1✮ t❡♠♦s ♥ ❡❧❡♠❡♥t♦s ❞✐s♣♦♥í✈❡✐s✱

♣❛r❛ ❛ ♣♦s✐çã♦ x2 t❡♠♦s ♥✲✶ ❡❧❡♠❡♥t♦s ❞✐s♣♦♥í✈❡✐s✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛té ❛

♣♦s✐çã♦xp✱ ❛ q✉❛❧ t❡r❡♠♦s ✭♥ ✲ ✭♣ ✲ ✶✮✮ ❡❧❡♠❡♥t♦s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ t❛❧ ♣♦s✐çã♦✳ P❡❧♦

♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✿

An,p = n.(n −1).· · · .(n −(p−1)) ❡✱ ❛♦ ♠✉❧t✐♣❧✐❝❛r ❡ss❛ ❡①♣r❡ssã♦ ♣♦r (n

−_p_)! (n−p)!✱

t❡r❡♠♦s✿

An,p=n.(n−1).· · · .(n−p+ 1).(n

−_p_)! (n−_p_)! =

n! (n−_p_)!

❊①❡♠♣❧♦✳ ❊♠ ✉♠❛ ♣r♦✈❛ ❞❡ ♥❛t❛çã♦✱ ❝♦♠♣❡t❡♠ ✽ ❛t❧❡t❛s ❞❡ ♣❛ís❡s ❞✐❢❡r❡♥t❡s✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ♦ ♣ó❞✐♦ ♣♦❞❡ s❡r ❢♦r♠❛❞♦❄

❙♦❧✉çã♦✳ ❆ ❢♦r♠❛çã♦ ❞♦ ♣ó❞✐♦ é ❝♦♠♣♦st♦ ❞❡ ✸ ❛t❧❡t❛s✱ ❞♦s q✉❛✐s ❡st❛rã♦ ♣r❡✲ s❡♥t❡s ♥♦ ❣r✉♣♦ ❞❡ ✽ ❝♦♠♣❡t✐❞♦r❡s✳ ❖❜s❡r✈❡ q✉❡ ❛ ♦r❞❡♠ ❞♦s ♥❛❞❛❞♦r❡s ♥♦ ♣ó❞✐♦ ❢❛③ ❞✐❢❡r❡♥ç❛✱ ♣♦✐s ✉♠ ♠❡s♠♦ ❛t❧❡t❛ ❛♦ t✐r❛r ❡♠ ♣r✐♠❡✐r♦ ♦✉ t❡r❝❡✐r♦ ❧✉❣❛r✱ r❡❝❡❜❡rá ♠❡❞❛❧❤❛ ❞✐❢❡r❡♥t❡ ❡ s✉❜✐rá ♥✉♠ ❛♥❞❛r ❞✐❢❡r❡♥t❡ ♥❛ ♣r❡♠✐❛çã♦✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s ✉♠ ❛rr❛♥❥♦ ❞❡ ✽ ❛t❧❡t❛s✱ ❞♦s q✉❛✐s ✸ ❢♦r♠❛rã♦ ♦ ♣ó❞✐♦✳ ❆ss✐♠✱

A8,3 = ₍₈−8!_3)! = 8! 5! =

8.7.6.5!

5! = 8.7.6 = 336 ♣ó❞✐♦s ❞✐❢❡r❡♥t❡s✳

❊①❡♠♣❧♦✳ ◆❛ ❜✐❜❧✐♦t❡❝❛ ❞❡ ✉♠❛ ❡s❝♦❧❛✱ ♦s ❧✐✈r♦s sã♦ r❡❣✐str❛❞♦s ❝♦♠ ✉♠ ❝ó❞✐❣♦ ❝♦♠♣♦st♦ ❞❡ ❞✉❛s ❧❡tr❛s ❞✐st✐♥t❛s ❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ três ❛❧❣❛r✐s♠♦s ❞✐st✐♥t♦s✳ P♦❞❡✲s❡ ✉s❛r q✉❛❧q✉❡r ✉♠❛ ❞❛s ✷✻ ❧❡tr❛s ❞♦ ❛❧❢❛❜❡t♦ ❡ q✉❛❧q✉❡r ❛❧❣❛r✐s♠♦ ❞❡ ✵ ❛ ✾✳ ❉❡ss❛ ❢♦r♠❛✱ ❝❛❧❝✉❧❡ ♦ ♥ú♠❡r♦ ❞❡ ❧✐✈r♦s q✉❡ ♣♦❞❡♠ s❡r ❝❛t❛❧♦❣❛❞♦s✳

❙♦❧✉çã♦✳ ❈♦♠♦ ❛s ❧❡tr❛s ❞❡✈❡♠ s❡r ❞✐st✐♥t❛s✱ ♦s ❝ó❞✐❣♦s ❆❇ ❡ ❇❆ ✭♣♦r ❡①❡♠♣❧♦✮ sã♦ r❡❢❡rê♥❝✐❛s ❛ ❧✐✈r♦s ❞✐st✐♥t♦s✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣❛r❛ ♦s ♥ú♠❡r♦s✱ ❛ s❡q✉ê♥❝✐❛ ✵✶✷ é ❞✐❢❡r❡♥t❡ ❞❛ s❡q✉ê♥❝✐❛ ✷✵✶✱ ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❧✐✈r♦s ❞✐❢❡r❡♥t❡s✳ ❊♥tã♦✱ ♣❛r❛ ❛s ❧❡tr❛s t❡♠♦s ✉♠ ❛rr❛♥❥♦ ❞❡ ✷✻ ❧❡tr❛s ♣❛r❛ ❡s❝♦❧❤❡r ❞✉❛s ❡ ♣❛r❛ ♦s ♥ú♠❡r♦s t❡♠♦s ✶✵ ❛❧❣❛r✐s♠♦s ♣❛r❛ ❡s❝♦❧❤❡r três✳ P❡❧♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ❞♦ ♣r✐♥❝í♣✐♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❛ ❝♦♥t❛❣❡♠✱ ♦s ❛rr❛♥❥♦s ❞❡✈❡♠ s❡r ♠✉❧t✐♣❧✐❝❛❞♦s✳

A26,2.A10,3 = ₍₂₆26!−_2)!. 10! (10−_3)! =

26! 24!.

10! 7! =

26.25.24! 24! .

10.9.8.7!

7! = 468000

P♦❞❡♠ s❡r ❝❛t❛❧♦❣❛❞♦s✱ ❡♥tã♦✱ ✹✻✽✵✵✵ ❧✐✈r♦s✳

❈♦♠❜✐♥❛çã♦ s✐♠♣❧❡s

◗✉❛♥❞♦ q✉❡r❡♠♦s ❢♦r♠❛r ✉♠ ❣r✉♣♦ ❝♦♠ ♣ ❡❧❡♠❡♥t♦s✱ ❡s❝♦❧❤✐❞♦s ❡♠ ✉♠ ✉♥✐✈❡rs♦ ❞❡ ♥ ❡❧❡♠❡♥t♦s✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ ♦r❞❡♠ ❞❡ss❛ ❡s❝♦❧❤❛ ♥ã♦ ✐♥✢✉❡♥❝✐❡ ♥♦ r❡s✉❧t❛❞♦✱ ❞✐③❡♠♦s q✉❡ ❡ss❡ ❣r✉♣♦ é ✉♠❛ ❝♦♠❜✐♥❛çã♦✳ P❛r❛ ❡ss❡ t✐♣♦ ❞❡ ❝♦♠❜✐♥❛çã♦ ✭s✐♠♣❧❡s✮✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ só ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ ✉♠❛ ú♥✐❝❛ ✈❡③✳ P♦❞❡♠♦s ❛ss♦❝✐❛r t❛♠❜é♠ ❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ♣ ❡❧❡♠❡♥t♦s ❛ ✉♠ ❝♦♥❥✉♥t♦✱ ♣♦✐s ❛ ♦r❞❡♠ ❞♦s ❡❧❡♠❡♥t♦s ❡♠ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ♥ã♦ ❢❛③ ❞✐❢❡r❡♥ç❛✳ P❛r❛ ❞❡t❡r♠✐♥❛r ❛ q✉❛♥t✐❞❛❞❡ t♦t❛❧ ❞❡ ❝♦♠❜✐♥❛çõ❡s ❞❛ ❡s❝♦❧❤❛ ❞❡ ♣ ❡❧❡♠❡♥t♦s ❞❡♥tr❡ ♦s ♥ ❡❧❡♠❡♥t♦s ❞✐s♣♦♥í✈❡✐s✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r✿

Cn,p=

n!

p!.(n₋p)!

❚❛♠❜é♠ ❝♦♥❤❡❝✐❞♦ ♣❡❧❛ ❢♦r♠❛ n p

✱ ❛s ❝♦♠❜✐♥❛çõ❡s t❛♠❜é♠ ♣♦❞❡♠ s❡r ❝♦♥s✐✲ ❞❡r❛❞❛s ❝♦♠♦ t♦❞♦s ♦s ❛rr❛♥❥♦s ✭r❡s✉❧t❛❞♦s ♦r❞❡♥❛❞♦s✮ ♥♦s q✉❛✐s só ♥♦s ✐♥t❡r❡ss❛♠

(18)

✶✳✶✳ ❆◆➪▲■❙❊ ❈❖▼❇■◆❆❚Ó❘■❆

❛q✉❡❧❡s q✉❡ ♦s ❡❧❡♠❡♥t♦s sã♦ ❞✐❢❡r❡♥t❡s✱ ❡ ♥ã♦ ❛ ♦r❞❡♠✳ ❉❡ss❛ ❢♦r♠❛✿

Cn,p = n_p

= An,p p! =

n.(n−₁₎_.₍_n.₂₎_.···_.₍_n−_p₊₁₎

p! =

n.(n−₁₎_.₍_n.₂₎_.···_.₍_n−_p₊₁₎

p! . (n−_p_)! (n−_p_)! =

n!

p!.(n−_p_)!✳

❉❡ ❢❛t♦✱ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❞❡ p ❡❧❡♠❡♥t♦s✱ t❡♠♦s p! ❢♦r♠❛s ❞❡ r❡♦r❞❡♥á✲❧♦s✳

❆ss✐♠✱ ❡♠ ✉♠ ❛rr❛♥❥♦✱ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❛♣❛r❡❝❡p!✈❡③❡s✳ ▲♦❣♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❥✉♥t♦

❞❡p ❡❧❡♠❡♥t♦s ✭♦✉ s❡❥❛✱ ❛ ♦r❞❡♠ ♥ã♦ ✐♠♣♦rt❛✮ é

An,p

p! .

❊①❡♠♣❧♦✳ ❯♠ ❤♦s♣✐t❛❧ ❞✐s♣õ❡ ❞❡ ✽ ♠é❞✐❝♦s ❡ ✶✺ ❡♥❢❡r♠❡✐r♦s✳ ❈❛❞❛ ❡q✉✐♣❡ ❞❡ ♣❧❛♥tã♦ é ❝♦♠♣♦st❛ ♣♦r ✷ ♠é❞✐❝♦s ❡ ✺ ❡♥❢❡r♠❡✐r♦s✳ ◗✉❛♥t❛s ❡q✉✐♣❡s ❞❡ ♣❧❛♥tã♦ ♣♦❞❡♠ s❡r ❢♦r♠❛❞❛s ❝♦♠ ♦s ❢✉♥❝✐♦♥ár✐♦s ❞✐s♣♦♥í✈❡✐s❄

❙♦❧✉çã♦✳ P❛r❛ ❛ ❢♦r♠❛çã♦ ❞❛ ❡q✉✐♣❡✱ ❛ ♦r❞❡♠ ❞❛ ❡s❝♦❧❤❛ ❞♦s ❢✉♥❝✐♦♥ár✐♦s ✭❞❡♥✲ tr♦ ❞❡ ✉♠❛ ♠❡s♠❛ ❝❛t❡❣♦r✐❛✮ ♥ã♦ ✐♠♣♦rt❛✱ ✉♠❛ ✈❡③ q✉❡ ❛ ❡q✉✐♣❡ ❢♦r♠❛❞❛ ❝♦♠ ♦s ♠é❞✐❝♦s ❆ ❡ ❇ s❡rá ❛ ♠❡s♠❛ ❡q✉✐♣❡ ❢♦r♠❛❞❛ ❝♦♠ ♦s ♠é❞✐❝♦s ❇ ❡ ❆✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ ♦s ❡♥❢❡r♠❡✐r♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡q✉✐♣❡s ❝♦♠ ♠é❞✐❝♦s ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ✽ ❡❧❡♠❡♥t♦s ♣❛r❛ ❡s❝♦❧❤❡r ✷ ❞❡❧❡s✱ ❡ ❛ q✉❛♥t✐✲ ❞❛❞❡ ❞❡ ❡q✉✐♣❡s ❝♦♠ ❡♥❢❡r♠❡✐r♦s ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ✶✺ ❡❧❡♠❡♥t♦s ♣❛r❛ ❡s❝♦❧❤❡r ✺ ❞❡❧❡s✳ ❖ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ✉♥❡ ❛s ❞✉❛s ❝❛t❡❣♦r✐❛s ♥✉♠❛ ♠❡s♠❛ ❡q✉✐♣❡ ♣❧❛♥t♦♥✐st❛✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡✿

C8,2.C15,5 = _2!_.₍₈8!−_2)!. 15! 5!.(15−_5)! =

8.7.6! 2.1.6!.

15.14.13.12.11.10!

5.4.3.2.1.10! = 84084

▲♦❣♦✱ ♦ ❤♦s♣✐t❛❧ ♣♦❞❡ ♠♦♥t❛r ✽✹ ✵✽✹ ❡q✉✐♣❡s ❞✐❢❡r❡♥t❡s ♣❛r❛ s❡✉ ♣❧❛♥tã♦✳ ❊①❡♠♣❧♦✳ ❯♠❛ ❡s❝♦❧❛ t❡♠✱ ❡♠ s❡✉ ❝♦r♣♦ ❞♦❝❡♥t❡✱ ✷✵ ♣r♦❢❡ss♦r❡s ❞♦s q✉❛✐s s❡rã♦ ❡s❝♦❧❤✐❞♦s ✺ ♠❡♠❜r♦s ♣❛r❛ ♦ ❝♦♥s❡❧❤♦ ❡s❝♦❧❛r✳ ❆ ❡s❝♦❧❤❛ s❡rá ❛tr❛✈és ❞❡ ✉♠❛ ❡❧❡✐çã♦ ♣❛r❛ ♣r❡s✐❞❡♥t❡ ❡ ✈✐❝❡✲♣r❡s✐❞❡♥t❡ ❞♦ ❝♦♥s❡❧❤♦ ❡♠ q✉❡ ♦s ❞♦✐s ♠❛✐s ✈♦t❛❞♦s ❛ss✉♠❡♠ ♦s ❝❛r❣♦s ✭♥❡ss❛ ♦r❞❡♠✮ ❡✱ ❛♣ós ♦ r❡s✉❧t❛❞♦✱ ✉♠❛ ♥♦✈❛ ❡❧❡✐çã♦ ♣❛r❛ ❡s❝♦❧❤❡r três s❡❝r❡tár✐♦s✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ♦ ❝♦♥s❡❧❤♦ ❡s❝♦❧❛r ♣♦❞❡ s❡r ♠♦♥t❛❞♦❄

❙♦❧✉çã♦✳ ❖ ❝♦♥s❡❧❤♦ ❡s❝♦❧❛r s❡rá ♠♦♥t❛❞♦ ❡♠ ❞✉❛s ❡t❛♣❛s✿ ❛ ♣r✐♠❡✐r❛ ❡s❝♦❧❤❡✲s❡ ✉♠ ♣r❡s✐❞❡♥t❡ ❡ ✉♠ ✈✐❝❡✲♣r❡s✐❞❡♥t❡✱ tr❛t❛✲s❡ ♣♦rt❛♥t♦ ❞❡ ✉♠ ❛rr❛♥❥♦✱ ✉♠❛ ✈❡③ q✉❡ ❛ ♦r❞❡♠ ❞❡ss❡s ♠❡♠❜r♦s ✐♠♣♦rt❛✱ ♣♦✐s ❞❡♣❡♥❞❡ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈♦t♦s❀ ❛ s❡❣✉♥❞❛ ❡s❝♦❧❤❡ ✸ s❡❝r❡tár✐♦s ❞❡♥tr❡ ♦s ❞♦❝❡♥t❡s q✉❡ s♦❜r❛r❛♠✱ ♥❡ss❛ ❡❧❡✐çã♦ ♥ã♦ ✐♠♣♦rt❛ ❛ ♦r❞❡♠✱ ♦ t❡r❝❡✐r♦ ❧✉❣❛r ❡ ♦ q✉✐♥t♦ ❧✉❣❛r t❡rã♦ ♦ ♠❡s♠♦ ❝❛r❣♦✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s ✉♠ ❛rr❛♥❥♦ ❞❡ ✷✵ ❞♦❝❡♥t❡s q✉❡ s❡rã♦ ❡s❝♦❧❤✐❞♦s ✷ ❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ✶✽ ❞♦❝❡♥t❡s ♣❛r❛ s❡r❡♠ ❡s❝♦❧❤✐❞♦s ✸✳ ❆ss✐♠✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✿

A20,2.C18,3 = ₍₂₀20!−_2)!. 18! 3!.(18−_3!) =

20.19.18! 18! .

18.17.16.15!

3.2.1.15! = 310080

❆ ❡s❝♦❧❛ t❡♠ ✸✶✵✵✽✵ ♠♦❞♦s ❞❡ ❡s❝♦❧❤❡r s❡✉ ❝♦♥s❡❧❤♦ ❡s❝♦❧❛r✳

❈♦♠❜✐♥❛çã♦ ❝♦♠ r❡♣❡t✐çã♦

❚❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ❝♦♠❜✐♥❛çã♦ ❝♦♠♣❧❡t❛✱ ❡ss❡ t✐♣♦ ❞❡ ❝♦♠❜✐♥❛çã♦ é ✉t✐❧✐③❛❞❛ q✉❛♥❞♦ q✉❡r❡♠♦s ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ♠❛♥❡✐r❛s ❡s❝♦❧❤❡r p ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ♦✉

♥ã♦✱ ❡♥tr❡n ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞❛❞♦s✳

P❛r❛ ❡♥t❡♥❞❡r♠♦s ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠❜✐♥❛çõ❡s ❝♦♠ r❡♣❡t✐çõ❡s✱ ♣♦✲ ❞❡♠♦s ❛ss♦❝✐❛r ❝❛❞❛ ✉♠ ❞♦s n ❡❧❡♠❡♥t♦s ❛ ✉♠❛ ✈❛r✐á✈❡❧ xi✱ i = 1, . . . , n✱ ❞❡ t❛❧

❢♦r♠❛ q✉❡xi ❞❡♥♦t❛ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ♦i✲és✐♠♦ ❡❧❡♠❡♥t♦ ❢♦✐ ❡s❝♦❧❤✐❞♦✳ ❈♦♠♦

q✉❡r❡♠♦s ❡s❝♦❧❤❡r p ❡❧❡♠❡♥t♦s✱ ❡st❛♠♦s ♣r♦❝✉r❛♥❞♦ ♣❡❧❛ q✉❛♥t✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s

♥ã♦✲♥❡❣❛t✐✈❛s ❞❡

(19)

✶✳✶✳ ❆◆➪▲■❙❊ ❈❖▼❇■◆❆❚Ó❘■❆

x1+x2+· · ·+xn=p.

P♦❞❡♠♦s ❡♥tã♦ r❡s♦❧✈❡r ❡st❡ ♣r♦❜❧❡♠❛ ❝♦❧♦❝❛♥❞♦ p sí♠❜♦❧♦s _•✱ r❡♣r❡s❡♥t❛♥❞♦ ♦

t♦t❛❧ ❞❡ ❡❧❡♠❡♥t♦s ❛ s❡r❡♠ ❡s❝♦❧❤✐❞♦s✱ ❡✱ ❡♠ s❡❣✉✐❞❛ ♣♦s✐❝✐♦♥❛r n ₋1 sí♠❜♦❧♦s _|

❡♥tr❡ ❛s ❜♦❧❛s• ♣❛r❛ ✐♥❞✐❝❛r ❛s q✉❛♥t✐❞❛❞❡s ❞❡x1, x2, . . . , xn✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♦

♥ú♠❡r♦ ❞❡ ❜♦❧❛s•❛♥t❡s ❞♦ ♣r✐♠❡✐r♦ sí♠❜♦❧♦|✭❝♦♥t❛♥❞♦ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✮

é ✐❣✉❛❧ ❛ x1✱ ♦ ♥ú♠❡r♦ ❞❡ ❜♦❧❛s ❡♥tr❡ ♦ ♣r✐♠❡✐r♦ sí♠❜♦❧♦ | ❡ ♦ s❡❣✉♥❞♦ sí♠❜♦❧♦ |

é ✐❣✉❛❧ ❛ x2✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ▲❡♠❜r❛♥❞♦✱ q✉❡ ❝♦♠♦ ♣r♦❝✉r❛♠♦s s♦❧✉çõ❡s

♥ã♦✲♥❡❣❛t✐✈❛s é ♣♦ssí✈❡❧ ♥ã♦ t❡r♠♦s ♥❡♥❤✉♠❛ ❜♦❧❛ ❡♥tr❡ ♦i₋1✲és✐♠♦ sí♠❜♦❧♦ _|❡ ♦

i✲és✐♠♦ sí♠❜♦❧♦ _|✱ ✐♥❞✐❝❛♥❞♦ ♥❡st❡ ❝❛s♦ q✉❡xi = 0✳

❆ss✐♠✱ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ❛s s♦❧✉çõ❡s ♣♦ssí✈❡✐s ❡ ❛s s❡q✉ê♥❝✐❛s ❞❡• ❡ |❞❡ ❝♦♠♣r✐♠❡♥t♦ n₋1 +p✱ ❝♦♠n₋1 sí♠❜♦❧♦s_| ❡ psí♠❜♦❧♦s _•✳

❖ t♦t❛❧ ❞❡ t❛✐s s❡q✉ê♥❝✐❛s é ♦ ♥ú♠❡r♦ ❞❡ ♣❡r♠✉t❛çõ❡s ❞❡ n₋1 +p ❡❧❡♠❡♥t♦s ❝♦♠ p❡ n₋1 r❡♣❡t✐çõ❡s✿

(n₋1 +p)!

p!(n₋1)! .

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦ t♦t❛❧ ❞❡ ❝♦♠❜✐♥❛çõ❡s ❝♦♠♣❧❡t❛s ❛tr❛✈és ❞❛ r❡❧❛çã♦✿

CRp n=C

p

n+p−₁ =

(n+p₋1)!

p!.(n₋1)!

❊①❡♠♣❧♦✳ ◗✉❛♥t❛s s♦❧✉çõ❡s ♥❛t✉r❛✐s ✭✐♥t❡✐r❛s ♥ã♦✲♥❡❣❛t✐✈❛s✮ t❡♠ ❛ ❡q✉❛çã♦ ① ✰ ② ✰ ③ ❂ ✽ ❄

❙♦❧✉çã♦✳ P♦r s❡ tr❛t❛r ❞❡ s♦❧✉çõ❡s ♥❛t✉r❛✐s✱ ♦ ♠❡♥♦r r❡s✉❧t❛❞♦ ♣♦ssí✈❡❧ é ♦ ✵ ❡ ♦ ♠❛✐♦r r❡s✉❧t❛❞♦ ♣♦ssí✈❡❧ é ♦ ✽✳ P♦ré♠✱ ❡♠ t♦❞♦ ❡ss❡ ✐♥t❡r✈❛❧♦ ❞❡ ✾ ♥ú♠❡r♦s✱ ❞❡✈❡♠♦s ❡s❝♦❧❤❡r ✸ ❞❡❧❡s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ s♦❧✉çã♦ s❡❥❛ s❛t✐s❢❡✐t❛ ❡✱ ❧❡♠❜r❛♥❞♦✱ q✉❡ ❛❧❣✉♠❛ ✐♥❝ó❣♥✐t❛ ♣♦❞❡ t❡r ♦ ♠❡s♠♦ ✈❛❧♦r ❞❡ ♦✉tr❛✳ P❛r❛ ✐ss♦✱ ✈❛♠♦s ❝❤❛♠❛r ❝❛❞❛ ✉♥✐❞❛❞❡ ❞♦ r❡s✉❧t❛❞♦ q✉❡ q✉❡r❡♠♦s ❞❡ ✉♠ sí♠❜♦❧♦ ✭❛q✉✐ r❡♣r❡s❡♥t❛r❡♠♦s ❝♦♠ ✉♠ •✮✱ ❡ ❝❛❞❛ ♦♣❡r❛❞♦r q✉❡ s❡♣❛r❛ ❛s s♦❧✉çõ❡s ✐♥❞✐✈✐❞✉❛✐s ❞❡ ♦✉tr♦ sí♠❜♦❧♦ ✭❛q✉✐

✉s❛r❡♠♦s |✮✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛s s♦❧✉çõ❡s✱ ❛tr❛✈és ❞♦s sí♠❜♦❧♦s✱ ❝♦♠♦ ♣♦r

❡①❡♠♣❧♦✿ • • • | •• | • • • s❡ tr❛t❛ ❞❛ s♦❧✉çã♦ ✭✸❀✷❀✸✮ ❡ ❛ s♦❧✉çã♦ • || • • • • • • •

tr❛t❛✲s❡ ❞❛ t❡r♥❛ ✭✶❀✵❀✼✮✳ ❊♥tã♦✱ ❝♦♠♦ ♣♦❞❡♠♦s r❡♣❡t✐r ♦s r❡s✉❧t❛❞♦s ❡♠ ♠❛✐s ❞❡ ✉♠❛ ✈❛r✐á✈❡❧✱

CR38 =C8+33 −₁ =C 3

10 = 3!.(810!−_1)! =

10.9.8.7!

3.2.1.7! = 120

❍á ✶✷✵ s♦❧✉çõ❡s ♥❛t✉r❛✐s ♣❛r❛ ❛ ❡q✉❛çã♦ ①✰②✰③ ❂ ✽✳

❊①❡♠♣❧♦✳ ❯♠ ❜✉✛❡t t❡♠ ✼ ♦♣çõ❡s ❞❡ s❛❧❛❞❛s✱ ❞♦s q✉❛✐s ♦ ❝❧✐❡♥t❡ ♣♦❞❡ ❡s❝♦❧❤❡r ✹ ♣♦rçõ❡s ♣❛r❛ s❡✉ ❛❧♠♦ç♦✳ ❉❡ q✉❛♥t❛s ♠❛♥❡✐r❛s ♦ ❝❧✐❡♥t❡ ♣♦❞❡ ♠♦♥t❛r s❡✉ ♣r❛t♦ ❞❡ s❛❧❛❞❛❄

❙♦❧✉çã♦✳ ◆❛❞❛ ✐♠♣❡❞❡ ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❝❧✐❡♥t❡ ♣♦❞❡ ❡s❝♦❧❤❡r ✶ ♣♦rçã♦ ❞❡ ✉♠ t✐♣♦ ❞❡ ✸ ♣♦rçõ❡s ❞❡ ♦✉tr♦ t✐♣♦✳ ❊♥tã♦✱ ✈❛♠♦s ❛ss♦❝✐❛r ❝❛❞❛ t✐♣♦ ❞❡ s❛❧❛❞❛ ❛ ✉♠❛ ✈❛r✐á✈❡❧an✱ s❛❜❡♥❞♦ q✉❡ ♦ t♦t❛❧ ❞❡ ♣♦rçõ❡s ❞❡ ❝❛❞❛ t✐♣♦ ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ✹✳ ❆ss✐♠✱

a1+a2+a3+a4+a5+a6+a7 = 4✳ ❯t✐❧✐③❛♥❞♦ sí♠❜♦❧♦s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❡①❡♠♣❧♦s

❞❡ s♦❧✉çã♦ ❝♦♠♦ • ||| • || •• | q✉❡ ❢♦r♠❛ ❛ s♦❧✉çã♦ ✭✶✱✵✱✵✱✶✱✵✱✷✱✵✮ q✉❡ ✐♥❞✐❝❛ q✉❡

♦ ❝❧✐❡♥t❡ ❡stá q✉❡r❡♥❞♦ ✶ ♣♦rçã♦ ❞❛ s❛❧❛❞❛ ✶✱ ✶ ♣♦rçã♦ ❞❛ s❛❧❛❞❛ ✹ ❡ ✷ ♣♦rçõ❡s ❞❛ s❛❧❛❞❛ ✻✳ ❊♥tã♦✱ ✉s❛♥❞♦ ❛s ❝♦♠❜✐♥❛çõ❡s q✉❡ ♣♦❞❡♠ t❡r ❡❧❡♠❡♥t♦s r❡♣❡t✐❞♦s✱

CR4

7 =C7+44 −₁ =C₁₀4 = 10! 4!.(10−_4)! =

10.9.8.7.6!

4.3.2.1.6! = 210

(20)

✶✳✷✳ P❘❖❇❆❇■▲■❉❆❉❊

▲♦❣♦✱ ♦ ❝❧✐❡♥t❡ t❡rá ✷✶✵ ♠♦❞♦s ❞❡ ♠♦♥t❛r s❡✉ ♣r❛t♦ ❞❡ s❛❧❛❞❛s✳

✶✳✷ Pr♦❜❛❜✐❧✐❞❛❞❡

✶✳✷✳✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s

❆ ♣r✐♠❡✐r❛ ♦❜r❛ q✉❡ tr❛t❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❝❤❛♠❛✲s❡ ❉❡ ▲✉❞♦ ❆❧❡❛❡✱ ❞❡ ●✐♦✲ ❧❛r♠♦ ❈❛r❞❛♥♦ ✭✶✺✵✶✲✶✺✼✻✮✱ q✉❡ tr❛t❛ ❞❡ ❥♦❣♦s ❞❡ ❛③❛r❀ ♣♦ré♠ ❡st❛ só ❢♦✐ ♣✉❜❧✐❝❛❞❛ ❡♠ ✶✻✻✸✳ ❯♠ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ❈❛r❞❛♥♦ ❢♦✐ ♦ ✧♣r♦❜❧❡♠❛ ❞♦s ♣♦♥t♦s✧✱ q✉❡ ❧♦❣♦ ❢♦✐ ♣r♦♣♦st♦ ❛ P❛s❝❛❧ ✭✶✻✷✸✲✶✻✻✷✮✱ q✉❡ ♦ ❧❡✈♦✉ ♣❛r❛ ❋❡r♠❛t ✭✶✻✵✶✲✶✻✻✺✮✳ ❆ ♣❛rt✐r ❞❛í✱ ❤♦✉✈❡ ✉♠❛ ✐♠♣♦rt❛♥t❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ❛♠❜♦s✱ q✉❡ ❢❡③ ❝♦♠ q✉❡ ❝❛❞❛ ✉♠ ❞❡❧❡s r❡s♦❧✈❡ss❡ ❛ s✉❛ ♠❛♥❡✐r❛✱ P❛s❝❛❧ ✉t✐❧✐③❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ❛r✐t♠ét✐❝♦ ✭q✉❡ ✜❝♦✉ ❢❛♠♦s♦ ❝♦♠♦ ❚r✐â♥❣✉❧♦ ❞❡ P❛s❝❛❧✮ ❡ ❡ss❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞❡✉ ❢✉♥❞❛♠❡♥t♦s à t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ♠♦❞❡r♥❛✳ ❈❤r✐st✐❛❛♥ ❍✉②❣❡♥s ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❛ ❞❛r ✉♠ tr❛t❛♠❡♥t♦ ❝✐❡♥tí✜❝♦✱ ♠❛s ❢♦✐ ❏❛❦♦❜ ❇❡r♥♦✉❧❧✐ ✭❝♦♠ ❛ ❆rt❡ ❞❛ ❈♦♥❥❡❝t✉r❛✮ ❡ ❆❜r❛❤❛♠ ❞❡ ▼♦✐✲ ✈r❡ ✭❝♦♠ ❛ ❉♦✉tr✐♥❛ ❞❛ Pr♦❜❛❜✐❧✐❞❛❞❡✮ q✉❡ r❡❛❧♠❡♥t❡ s❡ r❡❢❡r✐r❛♠ à Pr♦❜❛❜✐❧✐❞❛❞❡ ❝♦♠♦ s❡♥❞♦ ✉♠ r❛♠♦ ❞❛ ♠❛t❡♠át✐❝❛✳

✶✳✷✳✷ ❈❛❧❝✉❧❛♥❞♦ Pr♦❜❛❜✐❧✐❞❛❞❡s

P❛r❛ ❡♥t❡♥❞❡r ♦ q✉❡ é ♣r♦❜❛❜✐❧✐❞❛❞❡✱ ✈❛♠♦s ❞❡✜♥✐r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ s❡rã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡s ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛✿

· ❊①♣❡r✐♠❡♥t♦ ❛❧❡❛tór✐♦ é t♦❞♦ ❡①♣❡r✐♠❡♥t♦ ✭♦✉ ❢❡♥ô♠❡♥♦✮ q✉❡ ♣r♦❞✉③ r❡✲

s✉❧t❛❞♦s ✐♠♣r❡✈✐sí✈❡✐s✱ ❞❡♥tr❡ ♦s ♣♦ssí✈❡✐s✱ ♠❡s♠♦ q✉❛♥❞♦ r❡♣❡t✐❞♦ ❡♠ s❡♠❡❧❤❛♥t❡s ❝♦♥❞✐çõ❡s✱ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞♦ ❛❝❛s♦✳

· ❊s♣❛ç♦ ❛♠♦str❛❧ ✭Ω✮ é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦ssí✈❡✐s r❡s✉❧t❛❞♦s ❞❡ ✉♠

❡①♣❡r✐♠❡♥t♦ ❛❧❡❛tór✐♦✳

· ❊✈❡♥t♦ é t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❞♦ ❡s♣❛ç♦ ❛♠♦str❛❧ ❞❡ ✉♠ ❡①♣❡r✐♠❡♥t♦ ❛❧❡❛tór✐♦✳

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ✉♠ ❡✈❡♥t♦ ❆ ❞❡ ✉♠ ❡s♣❛ç♦ ❛♠♦str❛❧Ω✜♥✐t♦ ❡ ❡q✉✐♣r♦✈á✈❡❧✳ ❉❡✲

✜♥✐♠♦s Pr♦❜❛❜✐❧✐❞❛❞❡ ❞♦ ❡✈❡♥t♦ ❆ ✭P✭❆✮✮ ❝♦♠♦ s❡♥❞♦ ❛ r❛③ã♦ ❡♥tr❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ❆ ✭♥✭❆✮✮ ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ❞❡Ω ✭♥✭Ω✮✮✳

P(A) = n(A)

n(Ω)

◆♦t❡ q✉❡✱ s❡ ❆ é ✉♠ ❡✈❡♥t♦ q✉❛❧q✉❡r ❞❡ Ω✱ ❛♦ ❝♦♥s✐❞❡r❛r♠♦s ♦s ❝♦♥❥✉♥t♦s ∅✱ ❆

❡Ω✱ t❡♠♦s q✉❡✿

∅_⊂_A_⊂_Ω_⇒_n₍∅₎_≤_n₍_A₎_≤_n_(Ω)_⇒ n(∅) n(Ω) ≤

n(A)

n(Ω) ≤

n(Ω)

n(Ω) ⇒0≤P(A)≤1✳

❙❡ ❆ é ✉♠ ❡✈❡♥t♦ ✐♠♣♦ssí✈❡❧✱ t❡♠♦s P✭❆✮ ❂ ✵✱ ♣♦✐s✿ A=∅_⇒_P₍_A_{) =} n(∅) n(Ω) = 0✳

❙❡ ❆ é ✉♠ ❡✈❡♥t♦ ❝❡rt♦✱ t❡♠♦s P✭❆✮ ❂ ✶✱ ♣♦✐s✿ A= Ω_⇒P(A) = n_n(Ω)_(Ω) = 1✳

❊①❡♠♣❧♦✳ ❯♠ ❞❛❞♦ é ❧❛♥ç❛❞♦ ❞✉❛s ✈❡③❡s ❝♦♥s❡❝✉t✐✈❛s ❡ ♦ r❡s✉❧t❛❞♦ ❞❡ s✉❛ ❢❛❝❡ ✈♦❧t❛❞❛ ♣❛r❛ ❝✐♠❛ é ❛♥♦t❛❞♦✳ ◗✉❛❧ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❛ s♦♠❛ ❞♦s ✈❛❧♦r❡s ❛♥♦t❛❞♦s s❡r ♠❛✐♦r q✉❡ ✾❄

❙♦❧✉çã♦✳ P❛r❛ ❡ss❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ✭✐♥✐❝✐❛❧♠❡♥t❡✮ ♦ ❡s♣❛ç♦ ❛♠♦str❛❧ Ω✳ ❊♠ ❞♦✐s ❧❛♥ç❛♠❡♥t♦s ❝♦♥s❡❝✉t✐✈♦s✱ ♣♦❞❡♠♦s t❡r ❝♦♠♦ r❡s✉❧t❛❞♦s✿