What is the probability that when we roll four fair 6-sided dice, they won't all show the same number?

Ah...I see the flaw in my reasoning......try this.....

The probability that  they don't all show the same number is 1 minus the probability that they DO all show the same number.

But there are only 6 ways that they all show the same number.

So

1 - 6/1296 = 1290 / 1296 = 25/26 = about 99.5%

There are six things possible for the first die, five for the second, four for the third and three for the fourth.

And there are 6^4  = 1296 total possible outcomes for rolling four dice.

So

 [6 x 5 x 4 x 3 ] / 1296  =  360 / 1296 =  5/18 = about 28%

Sorry, this one is wrong. I get your reasoning but I dont know how to do it.

Chris, you made a little mistake when you simplified your fraction.