A die is rolled 5 times, find the probability of getting exactly two times even number and exactly two times number greater than 3. A dice is rolled 5 times, find the probability of getting exactly two times even number and exactly two times number greater than 3. The result should be ≈ 0.0707 = 7.07 %.
This question is a follow-up to this question.
Earlier i got hints and answers to separate it in 3 cases:
Case 1: Exactly two slots are even and greater than 3.
Choose 2 slots (out of 5) to be even and greater than 3: 5C25C2
Each of those 2 slots can be 4 or 6: 2×2
The remaining 3 slots all must be odd and not greater 3 (1 or 3): 2323

Case 2: Exactly 1 slot is even and greater than 3.
Choose 1 slot to be even and greater than 3: 
That slot can be 4 or 6: 2
Choose 1 slot out of remaining 4 to be even and not greater than 3: 
That slot must be 2: 1
Choose 1 slot of remaining 3 to be greater than 3 and odd: 
That slot must be 5: 1
Each of the remaining 2 slots must be odd and not greater than 3, that is 1 or 3: 2×2

Case 3: No slot is even and greater than 3
Choose 2 slots to be even and not greater than 3: 
Those 2 slots must be 2: 1
Choose 2 slots (out of the remaining 3) to be greater than 3 and odd: 
Those 2 slots must be 5: 1×1
The 1 remaining slot must be odd and not greater than 3, that is 1 or 3: 2

Then i have to multiply the numbers with each case together and sum up 3 cases, then divide by 6^5 to get 215/1944
How to calculate every single case before i sum up to get this result?? I don't know where i make mistake in calculation 
 A: There are three cases, according to the possible number of overlaps.  
We say a number is large if it is $>3$, small otherwise.
Case I: No overlap. There are two small even numbers and two large odd ones.  
the only large odd number is $5$, the only small even number is $2$.  Thus we just have to pick $2$ slots to be occupied by $2$ ($10$ ways to do it) 
and then $2$ slots to be occupied by $5$ ($3$ ways to do it) 
and then populate the fifth slot with either $1$ or $3$. ($2$ ways to do that).  Hence $$10^*3^*2=60$$
Case II.  Exactly one point of overlap.  
Pick the overlap slot ($5$ ways to do that).  
Populate it ($2$ ways to do that)
Pick the slot for the small even number ($4$ ways to do that)
Populate it ($1$ way to do that)
Pick the slot for the large odd number ($3$ ways to do that)
Populate it ($1$ way to do that)
Populate the remaining two slots ($4$ ways to do that)
Hence $$5^*2^*4^*1^*3^*1^*4=480$$
Case III  Two overlaps
Pick the overlap pair ($10$ ways to do that)
Populate it ($4$ ways to do that)
populate the remaining three slots ($8$ ways to do that)
Hence $$10^*4^*8=320$$
So (barring arithmetic error) the ways to get what you want  number $$60+480+320=860$$  As there are $6^5$ total combinations the probability is $$\frac {860}{6^5}=\frac {215}{1944}\sim .1106$$
