number of ways of forming a digit

Question

We have been given six digits 0,1,2,3,4,5. Now what is the no. of ways in which these six digits can be used to form a three digit number divisible by 3 provided that the repetition of digits is not allowed? I am fully aware that any number which is divisible by 3 must have the sum of its constituent digits a multiple of 3. So, how can we find out the possible number of ways?

Answer 1

The max of the sum of the digit number is $3+4+5=12.$

So, how about examining a number whose digit sum is either $3,6,9,12$ as the followings?

1) The $3$ case does not lead anything because $111$ is not allowed.

2) The $6$ case leads the set of $\{321\}$, which means $123, 132,\cdots.$

3) The $9$ case leads the set of $\{531\},\{432\}$.

Then?