Exercícios Resolvidos de Análise Combinatória e Probabilidade em L A T E X PDF

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Exercícios Resolvidos de Análise Combinatória e Probabilidade em LTEX A S. R. Santos e L. A. M. S. Junior Universidade Federal do Estado do Rio de Janeiro, Brasil 9 de setembro de 2008 Resumo Este texto contém exercícios resolvidos de Análise Combinatória e Probabilidade do livro do Morgado e do Deg...

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❊①❡r❝í❝✐♦s ❘❡s♦❧✈✐❞♦s ❞❡ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ❡ Pr♦❜❛❜✐❧✐❞❛❞❡ ❡♠ ▲❚❊❳ ❆

❙✳ ❘✳ ❙❛♥t♦s ❡ ▲✳ ❆✳ ▼✳ ❙✳ ❏✉♥✐♦r ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊st❛❞♦ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ❇r❛s✐❧ ✾ ❞❡ s❡t❡♠❜r♦ ❞❡ ✷✵✵✽

❘❡s✉♠♦ ❊st❡ t❡①t♦ ❝♦♥té♠ ❡①❡r❝í❝✐♦s r❡s♦❧✈✐❞♦s ❞❡ ❆♥á❧✐s❡ ❈♦♠❜✐♥❛tór✐❛ ❡ Pr♦❜❛❜✐❧✐❞❛❞❡ ❞♦ ❧✐✈r♦ ❞♦ ▼♦r❣❛❞♦ ❡ ❞♦ ❉❡❣r♦♦t✳

✶ ■♥tr♦❞✉çã♦ ❊st❡ t❡①t♦ s✉r❣✐✉ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♣r♦✈❡r ♦s ❞✐s❝❡♥t❡s ❞❡ ♠❛t❡r✐❛❧ ❝♦♠ ❡①❡r❝í✲ ❝✐♦s s♦❜r❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♥♦ ❝❛s♦ ❞✐s❝r❡t♦✳ ❆ t❡♦r✐❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞♦s ❡①❡r❝í❝✐♦s ♥ã♦ s❡ ❡♥❝♦♥tr❛ ❛q✉✐✳

✷ ❈♦♠❜✐♥❛çõ❡s ❡ P❡r♠✉t❛çõ❡s ✷✳✶

Pr✐♥❝í♣✐♦ ❞❛ ❆❞✐çã♦ ❡ ❞❛ ▼✉❧t✐♣❧✐❝❛çã♦

✶✳ ◗✉❛♥t❛s ♣❛❧❛✈r❛s ❝♦♥t❡♥❞♦ ✸ ❧❡tr❛s ❞✐❢❡r❡♥t❡s ♣♦❞❡♠ s❡r ❢♦r♠❛❞❛s ❝♦♠ ✉♠ ❛❧❢❛❜❡t♦ ❞❡ ✷✻ ❧❡tr❛s❄ ❘✿ ❆ ♣r✐♠❡✐r❛ ❧❡tr❛ ♣♦❞❡ s❡r s❡❧❡❝✐♦♥❛❞❛ ❞❡ ✷✻ ♠❛♥❡✐r❛s✳ ♣r✐♠❡✐r❛✱ ❛ s❡❣✉♥❞❛ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ ✷✺ ♠❛♥❡✐r❛s✳ ❣✉♥❞❛✱ ❛ t❡r❝❡✐r❛ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞❛ ❞❡ ✷✹ ♠❛♥❡✐r❛s✳ ❢♦r♠❛❞❛s

26 × 25 × 24 = 15600

❊s❝♦❧❤✐❞❛ ❛

❊s❝♦❧❤✐❞❛ ❛ s❡✲

❆ss✐♠✱ ♣♦❞❡♠ s❡r

♣❛❧❛✈r❛s ❝♦♠ ✸ ❧❡tr❛s ❞✐❢❡r❡♥t❡s✳

✷✳ ◗✉❛♥t♦s sã♦ ♦s ❣❛❜❛r✐t♦s ♣♦ssí✈❡✐s ❞❡ ✉♠ t❡st❡ ❞❡ ✶✵ q✉❡stõ❡s ❞❡ ♠ú❧t✐♣❧❛ ❡s❝♦❧❤❛✱ ❝♦♠ ❝✐♥❝♦ ❛❧t❡r♥❛t✐✈❛s ♣♦r q✉❡stã♦❄

5 × 5 × 5 × . . . × 5 = 510 = 9765625

❘✿

✸✳ ◗✉❛♥t♦s ✐♥t❡✐r♦s ❤á ❡♥tr❡ ✶✵✵✵ ❡ ✾✾✾✾ ❝✉❥♦s ❛❧❣❛rís♠♦s sã♦ ❞✐st✐♥t♦s❄

9 × 9 × 8 × 7 = 4536

❘✿

✹✳ ❉❡ q✉❛♥t♦s ♠♦❞♦s ♣♦❞❡♠ s❡r ❡s❝♦❧❤✐❞♦s ✉♠ ♣r❡s✐❞❡♥t❡ ❡ ✉♠ s❡❝r❡tár✐♦ ❞❡ ✉♠ ❝♦♥s❡❧❤♦ q✉❡ t❡♠ ✶✷ ♠❡♠❜r♦s❄

12 × 11 = 132

❘✿

✶

✺✳ ❉❡ q✉❛♥t♦s ♠♦❞♦s ✸ ♣❡ss♦❛s ♣♦❞❡♠ s❡♥t❛r✲s❡ ❡♠ ✺ ❝❛❞❡✐r❛s ❡♠ ✜❧❛❄ ❘✿5 × 4 × 3 = 60 ✻✳ ◗✉❛♥t♦s ♥ú♠❡r♦s ❞❡ q✉❛tr♦ ❞í❣✐t♦s sã♦ ♠❛✐♦r❡s ❞♦ q✉❡ ✷✹✵✵ ❡✿ ✭❛✮ tê♠ t♦❞♦s ♦s ❞í❣✐t♦s ❞✐❢❡r❡♥t❡s❀ ❙❡ ♦ ♥ú♠❡r♦ ❝♦♠❡ç❛r ♣♦r ✷ t❡r❡♠♦s ✉♠❛ ú♥✐❝❛ ♦♣çã♦ ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛rís♠♦✳ ❖ s❡❣✉♥❞♦ ♣♦❞❡ s❡r ♦ ✹ ♦✉ ♠❛✐♦r ✭✻ ♦♣çõ❡s✮✳ ❖ t❡r❝❡✐r♦ ♥ã♦ ♣♦❞❡ s❡r ✐❣✉❛❧ ❛♦ ♣r✐♠❡✐r♦ ❡ ❛♦ s❡❣✉♥❞♦ ❛❧❣❛rís♠♦s ✭✽ ♦♣çõ❡s✮ ❡ ♦ q✉❛rt♦ ♥ã♦ ♣♦❞❡ s❡r ✐❣✉❛❧ ❛♦s ✸ ❛❧❣❛rís♠♦s ❛♥t❡r✐♦r❡s ✭✼ ♦♣çõ❡s✮✳ ◆❡ss❡ ❝❛s♦✱ t❡r❡♠♦s 1 × 6 × 8 × 7 = 336 ♥ú♠❡r♦s✳ ❙❡ ♦ ♥ú♠❡r♦ ♥ã♦ ❝♦♠❡ç❛r ♣♦r ❞♦✐s✱ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛rís♠♦ ∈ / {0, 1, 2} ✭✼ ♦♣çõ❡s✮✳ ❖ s❡❣✉♥❞♦ ❛❧❣❛rís♠♦ ♥ã♦ ♣♦❞❡rá s❡r ✐❣✉❛❧ ❛♦ ♣r✐♠❡✐r♦ ✭✾ ♦♣çõ❡s✮✳ ❖ t❡r❝❡✐r♦ ❛❧❣❛rís♠♦ ♥ã♦ ♣♦❞❡rá s❡r ✐❣✉❛❧ ❛♦s ❞♦✐s ❛♥t❡r✐♦r❡s ✭✽ ♦♣çõ❡s✮ ❡ ♦ ú❧t✐♠♦ ♥ã♦ ♣♦❞❡rá s❡r ✐❣✉❛❧ ❛♦s ❛♥t❡r✐♦r❡s ✭✼ ♦♣çõ❡s✮✳ ◆❡ss❡ ❝❛s♦ t❡r❡♠♦s✿ 7 × 9 × 8 × 7 = 3528 ♥ú♠❡r♦s✳ ❆ss✐♠✱ sã♦ 336 + 3528 = 3864 ♥ú♠❡r♦s✳ ✭❜✮ ♥ã♦ tê♠ ❞í❣✐t♦s ✐❣✉❛✐s ❛ ✸✱ ✺ ♦✉ ✻❄ ❙❡ ♦ ♥ú♠❡r♦ ❝♦♠❡ç❛r ♣♦r ✷ t❡r❡♠♦s ✉♠❛ ú♥✐❝❛ ♦♣çã♦ ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛rís♠♦✳ ❖ s❡❣✉♥❞♦ ❛❧❣❛rís♠♦ ∈ / {0, 1, 2, 3, 5, 6} ✭✹ ♦♣çõ❡s✮✳ ❖s ❞♦✐s ú❧t✐♠♦s ❛❧❣❛rís♠♦s ∈ / {3, 5, 6} ✭✼ ♦♣çõ❡s ❝❛❞❛✮✳ ◆❡ss❡ ❝❛s♦ t❡r❡♠♦s✿ 1 × 4 × 7 × 7 = 196 ♥ú♠❡r♦s✳ ▼❛s ♦s ❞♦✐s ú❧t✐♠♦s ❛❧❣❛rís♠♦s ♥ã♦ ♣♦❞❡♠ s❡r ✐❣✉❛✐s ❛ ③❡r♦✱ q✉❛♥❞♦ ♦ ♣r✐♠❡✐r♦ ❢♦r ✐❣✉❛❧ ❛ ✷ ❡ ♦ s❡❣✉♥❞♦ ✐❣✉❛❧ ❛ ✹✱ ♣♦✐s ♦ ♥ú♠❡r♦ t❡♠ q✉❡ s❡r ♠❛✐♦r ❞♦ q✉❡ ✷✹✵✵✳ ▲♦❣♦✱ t❡r❡♠♦s 196 − 1 = 195 ♥ú♠❡r♦s ❙❡ ♦ ♥ú♠❡r♦ ♥ã♦ ❝♦♠❡ç❛r ♣♦r ❞♦✐s✱ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛rís♠♦ ∈ / {0, 1, 2, 3, 5, 6} ✭✹ ♦♣çõ❡s✮✳ ❖ s❡❣✉♥❞♦✱ t❡r❝❡✐r♦ ❡ q✉❛rt♦ ❛❧❣❛rís♠♦s ∈ / {3, 5, 6} ✭✼ ♦♣çõ❡s ❝❛❞❛✮✳ ◆❡ss❡ ❝❛s♦ t❡r❡♠♦s✿ 4 × 7 × 7 × 7 = 1372 ♥ú♠❡r♦s✳ ❆ss✐♠✱ sã♦ 195 + 1372 = 1567 ♥ú♠❡r♦s✳ ✭❝✮ tê♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❛ ❡ ❜ s✐♠✉❧tâ♥❡❛♠❡♥t❡❄ ❙❡ ♦ ♥ú♠❡r♦ ❝♦♠❡ç❛r ♣♦r ✷ t❡r❡♠♦s ✉♠❛ ú♥✐❝❛ ♦♣çã♦ ♣❛r❛ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛rís♠♦✳ ❖ s❡❣✉♥❞♦ ∈ / {0, 1, 2, 3, 5, 6} ✭✹ ♦♣çõ❡s✮✳ ❖ t❡r❝❡✐r♦ ∈ / {3, 5, 6} ❡ ♥ã♦ ♣♦❞❡ s❡r ✐❣✉❛❧ ❛♦ ♣r✐♠❡✐r♦ ❡ ❛♦ s❡❣✉♥❞♦ ❛❧❣❛rís♠♦s ✭✺ ♦♣çõ❡s✮ ❡ ♦ q✉❛rt♦ ∈ / {3, 5, 6} ❡ ♥ã♦ ♣♦❞❡ s❡r ✐❣✉❛❧ ❛♦s ✸ ❛❧❣❛rís♠♦s ❛♥t❡r✐♦r❡s ✭✹ ♦♣çõ❡s✮✳ ◆❡ss❡ ❝❛s♦✱ t❡r❡♠♦s 1×4×5×4 = 80 ♥ú♠❡r♦s✳ ❙❡ ♦ ♥ú♠❡r♦ ♥ã♦ ❝♦♠❡ç❛r ♣♦r ❞♦✐s✱ ♦ ♣r✐♠❡✐r♦ ❛❧❣❛rís♠♦ ∈ / {0, 1, 2, 3, 5, 6} ✭✹ ♦♣çõ❡s✮✳ ❖ s❡❣✉♥❞♦ ❛❧❣❛rís♠♦ ∈ / {3, 5, 6} ❡ ♥ã♦ ♣♦❞❡rá s❡r ✐❣✉❛❧ ❛♦ ♣r✐♠❡✐r♦ ✭✻ ♦♣çõ❡s✮✳ ❖ t❡r❝❡✐r♦ ❛❧❣❛rís♠♦ ∈ / {3, 5, 6} ❡ ♥ã♦ ♣♦❞❡rá s❡r ✐❣✉❛❧ ❛♦s ❞♦✐s ❛♥t❡r✐♦r❡s ✭✺ ♦♣çõ❡s✮ ❡ ♦ ú❧t✐♠♦ ∈ / {3, 5, 6} ❡ ♥ã♦ ♣♦❞❡rá s❡r ✐❣✉❛❧ ❛♦s ❛♥t❡r✐♦r❡s ✭✹ ♦♣çõ❡s✮✳ ◆❡ss❡ ❝❛s♦ t❡r❡♠♦s✿ 4 × 6 × 5 × 4 = 480 ♥ú♠❡r♦s✳ ❆ss✐♠✱ sã♦ 480 + 80 = 560 ♥ú♠❡r♦s✳

✷

✼✳ ❖ ❝♦♥❥✉♥t♦ ❆ ♣♦ss✉✐ ✹ ❡❧❡♠❡♥t♦s ❡ ♦ ❝♦♥❥✉♥t♦ ❇ ♣♦ss✉✐ ✼ ❡❧❡♠❡♥t♦s✳ ◗✉❛♥t❛s sã♦ ❛s ❢✉♥çõ❡s

f : A 7−→ B ❄

◗✉❛♥t❛s sã♦ ❛s ❢✉♥çõ❡s ✐♥❥❡t♦r❛s

f : A 7−→ B ❄ ❊♠ ✉♠❛ ❢✉♥çã♦✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❆ t❡♠ q✉❡ ❝♦rr❡s♣♦♥❞❡r ❛ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❇✳ P❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ ❆ t❡♠♦s ✼ ♦♣çõ❡s ♣❛r❛

7 × 7 × 7 × 7 = 74 = 2401

❝♦rr❡❧❛❝✐♦♥❛r ❝♦♠ ❇✳ ▲♦❣♦ t❡♠♦s ❊♠ ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t♦r❛✱

❢✉♥çõ❡s✳

❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❆ t❡♠ q✉❡ ❝♦r✲

r❡s♣♦♥❞❡r ❛ ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ ❇✱ ♦✉ s❡❥❛✱ ♥ã♦ ♣♦❞❡ t❡r ❞♦✐s ❡❧❡♠❡♥t♦s ❞❡ ❆ s❡ ❝♦rr❡❧❛❝✐♦♥❛♥❞♦ ❝♦♠ ♦ ♠❡s♠♦ ❡❧❡♠❡♥t♦ ❞❡ ❇✳ P❛r❛ ♦ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❞❡ ❆ t❡♠♦s ✼ ♦♣çõ❡s ♣❛r❛ ❝♦rr❡❧❛❝✐♦♥❛r ❝♦♠ ❇✳ ❊s❝♦✲ ❧❤✐❞♦ ❡st❡✱ ♣❛r❛ ♦ s❡❣✉♥❞♦ ❡❧❡♠❡♥t♦ ❞❡ ❆ t❡♠♦s ✻ ♦♣çõ❡s ♣❛r❛ ❝♦rr❡❧❛❝✐♦♥❛r ❝♦♠ ❇ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ▲♦❣♦ t❡♠♦s

7×6×5×4 = 840 ❢✉♥çõ❡s ✐♥❥❡t♦r❛s✳

✽✳ ◗✉❛♥t♦s ❞✐✈✐s♦r❡s ♥❛t✉r❛✐s ♣♦ss✉✐ ♦ ♥ú♠❡r♦ ✸✻✵❄ ◗✉❛♥t♦s sã♦ ♣❛r❡s❄

360 = 23 ×32 ×51 ✳ ❖s ❞✐✈✐s♦r❡s ♥❛t✉r❛✐s ❞❡ ✸✻✵ sã♦ ❞❛ ❢♦r♠❛ 2a ×3b ×5c ✱ ❝♦♠ a = 0, 1, 2, 3; b = 0, 1, 2 ❡ c = 0, 1✳ ▲♦❣♦✱ sã♦ 4×3×2 = 24 ♦s ❞✐✈✐s♦r❡s ♥❛t✉r❛✐s ❞❡ ✸✻✵✳ ❊ 3 × 3 × 2 = 18 sã♦ ♣❛r❡s✳

❘✿

✾✳ ◗✉❛♥t♦s sã♦ ♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❞❡ ✹ ❞í❣✐t♦s q✉❡ ♣♦ss✉❡♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❞í❣✐t♦s ✐❣✉❛✐s❄ ❘✿ ❖s ♥ú♠❡r♦s ❝♦♠ ✹ ❞í❣✐t♦s sã♦

9 × 10 × 10 × 10 = 9000✳

◆ú♠❡r♦s ❝♦♠ ✹ ❞í❣✐t♦s ❞✐st✐♥t♦s✿

9 × 9 × 8 × 7 = 4536✳

❆❣♦r❛✱ ❜❛st❛ ❞✐♠✐♥✉✐r✿

9000 − 4536 = 4464✳

✶✵✳ ◗✉❛♥t♦s s✉❜❝♦♥❥✉♥t♦s ♣♦ss✉✐ ✉♠ ❝♦♥❥✉♥t♦ q✉❡ t❡♠ ♥ ❡❧❡♠❡♥t♦s❄ ❘✿ ❯♠ ❝♦♥❥✉♥t♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♥❛ ❢♦r♠❛ ❖ ♣r✐♠❡✐r♦ ❡❧❡♠❡♥t♦ ❞❡ ❆✱

a1 ✱

A = {a1 + a2 + . . . + an }✳

♣♦❞❡ ♣❡rt❡♥❝❡r ❛♦ s✉❜❝♦♥❥✉♥t♦ ♦✉ ♥ã♦✳ ❖

s❡❣✉♥❞♦ t❛♠❜é♠✳ ❊ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❛té ♦ ♥✲és✐♠♦ ❡❧❡♠❡♥t♦✳ ▲♦❣♦✱ ♦ ♥ú♠❡r♦ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ ❆ é

2 × 2 × . . . × 2 = 2n ✳

✶✶✳ ❉❡ q✉❛♥t♦s ♠♦❞♦s ♣♦❞❡♠♦s ❛rr✉♠❛r ♦✐t♦ t♦rr❡s ✐❣✉❛✐s ❡♠ ✉♠ t❛❜✉❧❡✐r♦ ❞❡ ①❛❞r❡③

(8 × 8)

❞❡ ♠♦❞♦ q✉❡ ♥ã♦ ❤❛ ❥❛ ❞✉❛s t♦rr❡s ♥❛ ♠❡s♠❛ ❧✐♥❤❛ ♥❡♠

♥❛ ♠❡s♠❛ ❝♦❧✉♥❛❄ ❘✿ ◆❛ ♣r✐♠❡✐r❛ ❝♦❧✉♥❛ t❡♠♦s ✽ ♠♦❞♦s ❞❡ ❝♦❧♦❝❛r ❛ t♦rr❡✳ ◆❛ s❡❣✉♥❞❛✱ ✼✳ ◆❛ t❡r❝❡✐r❛✱ ✻ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ▲♦❣♦✱ t❡♠♦s

8 × 7 × . . . × 1 = 8! =

40320✳ ✶✷✳ ❊♠ ✉♠❛ ❜❛♥❝❛ ❤á ✺ ❡①❡♠♣❧❛r❡s ✐❣✉❛✐s ❞❛ r❡✈✐st❛ ❆✱ ✻ ❡①❡♠♣❧❛r❡s ❞❛ r❡✈✐st❛ ❇ ❡ ✶✵ ❡①❡♠♣❧❛r❡s ✐❣✉❛✐s ❞❛ r❡✈✐st❛ ❈✳ ◗✉❛♥t❛s ❝♦❧❡çõ❡s ♥ã♦ ✈❛③✐❛s ❞❡ r❡✈✐st❛s ❞❡ss❛ ❜❛♥❝❛ é ♣♦ssí✈❡❧ ❢♦r♠❛r❄ ❘✿

Ω = {(a, b, c) : a = ♥ú♠❡r♦s }

❞❡ r❡✈✐st❛s ❆

, b = ♥ú♠❡r♦

❞❡ r❡✈✐st❛s ❇ ❡

c=

♥ú♠❡r♦ ❞❡ r❡✈✐st❛s ❈

#Ω = 6 × 7 × 11 = 462✳

P♦ré♠✱ ♦ t❡r♥♦

(0, 0, 0)♥ã♦

♥ú♠❡r♦ ❞❡ ❝♦❧❡çõ❡s ♥ã♦ ✈❛③✐❛s s❡rá ✹✻✷✲✶❂✹✻✶✳

✸

♥♦s s❡r✈❡✳

▲♦❣♦✱ ♦

✶✸✳ ❉❡ ✉♠ ❜❛r❛❧❤♦ ❝♦♠✉♠ ✭✺✷ ❝❛rt❛s✮ s❛❝❛♠✲s❡ s✉❝❡ss✐✈❛♠❡♥t❡ ❡ s❡♠ r❡♣♦s✐çã♦ três ❝❛rt❛s✳ ◗✉❛♥t❛s sã♦ ❛s ❡①tr❛çõ❡s ♥❛s q✉❛✐s ❛ ♣r✐♠❡✐r❛ ❝❛rt❛ é ❞❡ ❝♦♣❛s✱ ❛ s❡❣✉♥❞❛ é ✉♠ r❡✐ ❡ ❛ t❡r❝❡✐r❛ ♥ã♦ é ✉♠❛ ❞❛♠❛❄ ❘✿ ❱❛♠♦s ❞✐✈✐❞✐r ♦ ♣r♦❜❧❡♠❛ ❡♠ três ♣❛rt❡s✳ Pr✐♠❡✐r❛ ♣❛rt❡✿ ◗✉❛♥t❛s sã♦ ❛s ❡①tr❛çõ❡s ♥❛s q✉❛✐s ❛ ♣r✐♠❡✐r❛ ❝❛rt❛ é ❞❡ ❝♦♣❛s ❡ ♥ã♦ é ♦ r❡✐ ❞❡ ❝♦♣❛s ❡ ♥ã♦ é ❛ ❞❛♠❛ ❞❡ ❝♦♣❛s ❡ ❛ s❡❣✉♥❞❛ é ✉♠ r❡✐ ❡ ❛ t❡r❝❡✐r❛ ♥ã♦ é ✉♠❛ ❞❛♠❛❄

11 × 4 × 46 = 2024

❙❡❣✉♥❞❛ ♣❛rt❡✿ ◗✉❛♥t❛s sã♦ ❛s ❡①tr❛çõ❡s ♥❛s q✉❛✐s ❛ ♣r✐♠❡✐r❛ ❝❛rt❛ é ♦ r❡✐ ❞❡ ❝♦♣❛s ❡ ❛ s❡❣✉♥❞❛ é ✉♠ r❡✐ ❡ ❛ t❡r❝❡✐r❛ ♥ã♦ é ✉♠❛ ❞❛♠❛❄

1×3×46 = 138

❚❡r❝❡✐r❛ ♣❛rt❡✿ ◗✉❛♥t❛s sã♦ ❛s ❡①tr❛çõ❡s ♥❛s q✉❛✐s ❛ ♣r✐♠❡✐r❛ ❝❛rt❛ é ❛ ❞❛♠❛ ❞❡ ❝♦♣❛s ❡ ❛ s❡❣✉♥❞❛ é ✉♠ r❡✐ ❡ ❛ t❡r❝❡✐r❛ ♥ã♦ é ✉♠❛ ❞❛♠❛❄

1 × 4 × 47 = 188 ▲♦❣♦✱ ❙ã♦

2024 + 138 + 188 = 2350

✹

❡①tr❛çõ❡s✳...