You have five choices for the first number.
The second number is also a choice from those five numbers, but it cannot be the same as the first, so really there are just four choices for the second number.
If there are five choices for the first number and four for the second, then we multiply these together to find out how many two-digit permutations we have (twenty).
For a three digit permutation, you can see there is one less choice for the third digit as the second digit. Since the third digit comes after one of the twenty two-digit permutations, we multiply again to get the number of three-digit permutations.
Hope this helps,
Stephen La Rocque.