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09 análise combinatória - parte ii (fatorial)

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09 análise combinatória - parte ii (fatorial)

  1. 1. Análise Combinatória – Parte IISoraya Mara Menezes de Souza
  2. 2. Análise Combinatória – Parte IIFatorialSendo n um número inteiro maior que 1,define-se fatorial de n como o produto dos nnúmeros naturais consecutivos de n a 1. n! = n ( n − 1) ( n − 2) ( n − 3) ... 3 ⋅ 2 ⋅ 1 sendo n ∈ Ν e n > 16! =6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 6! = 6 ⋅ 5!5! =5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 6! = 6 ⋅ 5 ⋅ 4!7! =7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1
  3. 3. Análise Combinatória – Parte IIFatorialCalcule:8! 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 40.320 = = = 3365! 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 120 ou8! 8 ⋅ 7 ⋅ 6 ⋅ 5! 8 ⋅7 ⋅6 = = = 3365! 5! 1
  4. 4. Análise Combinatória – Parte IIFatorialCalcule: 8! 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4! 1680 = = = 8404! ⋅ 2! 4! ⋅ 2 ⋅ 1 25! ⋅ 4! 5! ⋅ 4 ⋅ 3! 4 2 = = =3! ⋅ 6! 3! ⋅ 6 ⋅ 5! 6 3
  5. 5. Análise Combinatória – Parte IIFatorialResolva as equações:x! = 15 ( x − 1)! ( n − 2)! = 720x ( x − 1)! = 15 ( x − 1)! ( n − 2)! = 6! 15 ( x − 1)!x = n −2 = 6 ( x − 1)!x = 15 n =8
  6. 6. Análise Combinatória – Parte IIFatorialResolva as equações: x!( n − 2)! = 2 ( n − 4)! = 30 ( x − 2)!( n − 2)(n − 3)(n − 4)! = 2( n − 4)! x ( x − 1)( x − 2)!( n − 2)(n − 3) = 2( n − 4)! = 30 ( n − 4)! ( x − 2)!( n − 2)(n − 3) = 2 x ( x − 1) − 30 = 0n ² − 5n + 6 = 2 x ² − x − 30 = 0n ² − 5n + 4 = 0 x = 6 ou x = −5n = 4 ou n = 1 x =6 n =4
  7. 7. Análise Combinatória – Parte IIFatorialSimplifique as expressões: n! n! − ( n + 1)!( n − 1)! n!n( n − 1)! n! − (n + 1)n! n! (1 − n − 1) =n = = −n( n − 1)! n! n! ( n + 2)! ( n + 2)(n + 1) n( n − 1)! n(n + 2)(n + 1)   ( n − 1)! ( n − 1)!
  8. 8. Análise Combinatória – Parte IISoraya Mara Menezes de Souza

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