10! is a very large number, so I'm not going to write out all 3628800 possibilities!
This is one way of ordering numbers without repeating digits however:
For the first digit, choose the smallest number: 0
Thus the first number to write down is 0123456789.
Now for the next number, you back up:
Thus the second number is 0123456798.
Now back up again: we've gone through all the possibilities that start with 01234567, so we look at the eigth digit choices. They were 7, 8, and 9 and we chose 7, so now choose the "next" value: 8.
That leaves 7 and 9 for the ninth digit, so choose the smaller: 7.
Leaving 9 for the last digit, and so the next number is 0123456879.
Now back up and choose the next possibility for the ninth digit: 9.
This leaves 7 for the last digit, so the next number is 0123456897.
Again we back up to the eight digit and choose the next smallest number we haven't used from its set. Since we used 7 and 8 already, we use 9 this time.
Following the same steps, the next two numbers are 0123456978 and 0123456987.
In summary, we are "holding" the beginning constant and changing the endings until we've exhausted all the permutations of numbers that start with that beginning. Whenever we change a digit, we start a new sequence for all the digits its right.
Here's a complete but smaller example. What are all the possible arrangements of ABCD that don't repeat letters?
and as you observed, this is 4! = 24 permutations.
Hope this helps,
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.