If you are just using the digits from 1 to 6, the answer would be 6*5*4*3*2*1 = 720, because you have 6 choices for the units digit, and then 5 choices left for the tens, and then 4 choices left for the hundreds and so on. Similarly, if all the digits from 1 to 9 were allowed (with no digit repeated), then the answer would be 9*8*7*6*5*4 = 60480.

The tricky part comes when the digit 0 is allowed: suppose that you wanted to form a 6-digit number using digits from 0,1,...,9 without repeating a digit. Then 0 could not be the leftmost digit. So, starting from the left, you would have 9 choices for the leftmost digit (not allowing 0), then 9 choices again for the next digit (0 is now allowable but not the previous digit), then 8 choices for the next digit and so on, for a result of 9*9*8*7*6*5 = 136080.