Probability problem with a die I've been practicing probability problems lately and I came to this problem

A number is formed in the following way. You throw a six-sided
  die until you get a 6 or until you have thrown it three times at the
  most. A sequence of dice throws form either one, two or three-digit
  numbers. How many distinct numbers can be formed as a result of
  this experiment?

I thought about solving the problem this way:
First if the die is thrown and if it lands on a 6 that's the first number. 
Other numbers are:
16
26 36 46 56.
Now the next numbers are: 123, 124, 125... and so on. 
Is this right and if it is how do you approach this problem faster. It would take too long to solve it this way.

For case 3: Not $6$, Not $6$, Any value.  (Then use the multiplication principle).
