I am reading the following problem:

> 
> The digits $1$,$2$,$3$,$4$,$5$ are randomly arranged to form a three
> digit number (digits are not repeated). What is the probability that
> the number is even and greater than $500$?
> 
> 
> 

My solution:  

There are $5\cdot 4 \cdot 3$ ways to form the number in total.  

In order to be greater than $500$ the first digit must be $5$ and in order to be even the last digit must be either $2$ or $4$. So there is $1$ way to select the first digit, $2$ ways to select the last and $2$ ways to select the middle one (either $3$ or $4$) i.e. $2\cdot 2\cdot 1$ = $4$ ways  

Therefore the probability is: $\frac{4}{60}=\frac{1}{15}$

But the solution mentions $\frac{1}{10}$.  

What am I doing wrong here?