Question

комбинаторика ... не могу решить.... помогите

Answer 1

$1) x geq 2\C_{x-2}^{x-4}= frac{(x-2)!}{(x-4)!*2!} =(x-3)(x-2)/2\ (x-3)(x-2)=4x^2-14x-6\ x^2-5x+6 = 4x^2-14x-6\ 3x^2-9x-12=0\ x^2-3x-4=0\ D=9+16=25\ x = (3+5)/2=4$

$2) x geq 2\2C_{x-1}^2= frac{2(x-1)!}{(x-3)!2!} =(x-2)(x-1)\ 2C_x^{x-2} = frac{2x!}{(x-2)!2!} =(x-1)x\ (x-1)(x-2) + (x-1)x = (x-1)(x+1)\ (x-1)(x-2+x-x-1)=0\ x_1 = 1\ x-3=0\ x_2 =3$

Ответ: x = 3

$3)x geq 0;y geq 0\ C_x^y= frac{x!}{y!(x-y)!} = C_x^{y+2}= frac{x!}{(y+2)!(x-y-2)!} \ y!(x-y)!=(y+2)!(x-y-2)!\ (x-y-1)(x-y) = (y+1)(y+2)\ \ C_x^2= frac{(x-1)x}{2} =66\ x(x-1) = 132\ x^2 - x - 132 = 0\ D = 1 + 528 = 529 = 23^2\ x = (1+23)/2 = 12\ \ (12-y-1)(12-y)=(y+1)(y+2)\ 132-23y+y^2=y^2+3y+2\ 26y=130\ y=5$
Ответ: (12;5)