I was reading a book about inequalities, in that I found that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}>1\tag{1}$$ is a symmetric inequality in $a$,$b$,$c$. But if I change the order from $(a,b,c)$ to $(a,c,b)$, I am not getting the same inequality which is the basic definition of symmetric inequality. What am I doing wrong?
Here is my attempt: After changing the order to $(a,c,b)$, we get $$\frac{a}{c}+\frac{c}{b}+\frac{b}{a}>1$$ this is clearly not same as $(1)$.
Here is the excerpt from the book which I am studying: 
It's not symmetric inequality because the permutation $(a,b,c)\rightarrow(a,c,b)$ gives another inequality: $$\frac{a}{c}+\frac{b}{a}+\frac{c}{b}>1.$$
By the way, the inequality $$a+b+c>abc$$ is symmetric.