If x+y+z = 9; then what is the probability of having x = y = z = 3 (x, y, z are positive integers)

Question

If
x(x+y) = y(y+z) = z(z+x)
and
x,y,z are positive integers.
Then is it definitely true that x=y=z

Answer 1

Total number of cases will be ::

x + y = z = 6 { x>=0 ,y>=0 ,z>=0 }

(6+3-1)
C (3-1) =====

8
C =================28
2

hence the probaility of x=y=z=3 is 1/28

Answer 2

Since there are 28 options available and only one is favourable
Therefore Probability = 1/28