Mastering Combinatorics: Calculating Probabilities for Liar's Dice

[Hypercubes]

Probably the only area of math that really confuses me. I'm trying to calculate some probabilities for Liar's Dice. Essentially, the probabilities that a certain number of faces will appear when five dice are rolled, with one being a wildcard. If I try a specific combinatoric approach, for 5's for example, I get this. The numbers on each space represent the possible values:

Permutations that produce 0 5's: (2, 3, 4, 6)^5=4^5=1024
Permutations that produce 1 5: (1, 5)([2, 3, 4, 6]^4)=512
etc.

Already a problem emerges, as the chance of getting one 5 is higher than that of getting no 5's. If continued, it keeps dividing by two.However, if I use a general approach and ignore the wildcard property of ones, I get a (seemingly) correct answer, as when graphed it produces the familiar bell curve:
Permutations that produce 0 of anything: 5!=120 (don't quite understand the logic of this one)
Permutations that produce 1 of anything: (*)([2, 3, 4, 6]^4)=256
After that I can't figure out what I did, but here are the results:
120
256
320
80
20
1

What is the correct approach to this problem, and how to convert to probabilities?

Thanks in advance.

 

[JCVD]

When you calculate how many ways you can get zero 1s or 5s you count as though you are rolling the dice one at a time and order of results matter (ie, rolling 2,2,4,4,4 is different from rolling 4,4,4,2,2). This is fine as long as you are consistent throughout, but when you count how many ways you can get exactly one 1 or 5, you don't take into account where that 1 or 5 appears in the order of dice rolled. Since there are five possible places, you should multiply 512 by 5 to get 2560.