What is the probability of getting 1,2,3,4 when 5 dice are thrown and

[MilenaMath]

What is the probability of getting 1,2,3,4 when 5 dice are thrown and one of them is not 5.(the order of 1,2,3,4 doesn't matter)
I am really confused:
So I computed chances of getting correct numbers that is 1/6^4 times 5/6,then I multiply this by 4 times 5! Am I right?

 

[haruspex]

What is the probability of getting 1,2,3,4 when 5 dice are thrown and one of them is not 5.(the order of 1,2,3,4 doesn't matter)

Your question is unclear. Do you mean at least one each of 1, 2, 3, 4? Exactly one of each?
When you say one of them is not five, do you mean that a particular one is not five, that they are not all five, or that none of them are five?

 

[MilenaMath]

I need to get exactly 1,2,3,4(order doesn't matter) and the fifth dice can be whatever number but 5

 

[haruspex]

Sorry, still not clear. If you get 4, 2, 2, 3, 1, is that ok or not?

 

[Ray Vickson]

I need to get exactly 1,2,3,4(order doesn't matter) and the fifth dice can be whatever number but 5

From this description, it seems the event you want is
A = {1,1,2,3,4} or {1,2,2,3,4} or {1,2,3,3,4} or {1,2,3,4,4} or {1,2,3,4,6}, but without regard for order. Is that correct? If so, you can use a multinomial distribution to find
P{A} = p(2,1,1,1,0,0) + p(1,2,1,1,0,0) + ... + p(1,1,1,1,0,1), where these are the occupancy numbers in each of the 6 classes 1--6.

RGV