Question

(Комбинаторика)
Решите систему уравнений.

Answer 1

n! = 1*2*....*n

n ∈ N

a>b

Aₐᵇ = a! / (a - b)!

Cₐᵇ = a! / (a - b)!b!

Aₓⁿ⁻³ : Aₓⁿ⁻² = x!/(x - n + 3)! : x!/(x - n + 2)! =  x!/(x - n + 3)! * (x - n + 2)!/x! = 1 / (x - n + 3)

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(x - n + 3)! = 1*2*.......*(x - n + 2)(x - n + 3)

(x - n + 2)! = 1*2*.......*(x - n + 2)

(x - n + 2)! /  (x - n + 3)!  = (x - n + 3)

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Cₓⁿ⁻³ : Cₓⁿ⁻² = x!/(x - n + 3)!(n - 3)! : x!/(x - n + 2)!(n - 2)! =  x!/(x - n + 3)!(n - 3)! * (n -2)!(x - n + 2)!/x! = (n - 2) / (x - n + 3)

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(n - 2)! = 1*2*....*(n - 4)(n - 3)(n - 2)

(n - 3)! = 1*2*....*(n - 4)(n - 3)

(n - 2)! / (n - 3)! = n - 2

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1/( x - n + 3) = 1/8

(n - 2)/(x - n + 3) = 5/8

(n - 2) / 8 = 5/8

n - 2 = 5

n = 7

x - n + 3 = x - 7 + 3 = x - 4 = 8

x = 12