Joint PMF (calculating probabilities)

[thumper88]

So I have a problem that says four fair dice are rolled. So that tells me 6⁴ sample space.
X is the number of fives observed and Y is the number of sixes observed.
I created a table

PX,Y(x,y)	y=0	y=1	y=2	y=3	y=4	PX(x)
x=0
x=1					0
x=2				0	0
x=3			0	0	0
x=4		0	0	0	0
PY(y)

I understand the method of calculating the PMF, I'm having counting issues, calculating the probability for each part. I believe the zeros are placed correctly? If 4 fives are observed you can't have more than 0 sixes observed and vice versa.
For example, how do I calculate the probability of 2 fives observed and 2 sixes observed? Please help, thanks.

 

[pka]

So I have a problem that says four fair dice are rolled. So that tells me 6⁴ sample space.
X is the number of fives observed and Y is the number of sixes observed.

LaTeX is down so a fancy answer is out of the question .
Let #(x=m,y=n) mean the number of 4-tuples in which there are m fives and n sixes.
#(x=0,y=0)=4⁴ ,

#(x=1,y=2)=(4!/2)(4)=2(4!).

#(x=2,y=2)=6

You try the rest.