A dice 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 %. but i can't get it
 A: Think of $5$ rolls as an ordered sequence of $5$ slots to put numbers 1-6 in.
Case 1: Exactly two slots are even and greater than 3.


*

*Choose 2 slots (out of 5) to be even and greater than 3: ${}_5C_{2}$

*Each of those 2 slots can be 4 or 6: $2\times 2$

*The remaining 3 slots all must be odd and not greater 3 (1 or 3): $2^3$


Case 2: Exactly 1 slot is even and greater than 3.


*

*Choose $1$ slot to be even and greater than 3: ${}_5C_1$

*That slot can be 4 or 6: $2$

*Choose 1 slot out of remaining 4 to be even and not greater than 3: ${}_4C_1$

*That slot must be 2: $1$

*Choose 1 slot of remaining 3 to be greater than 3 and odd: ${}_3C_1$

*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: $2\times 2$


Case 3: No slot is even and greater than 3


*

*Choose 2 slots to be even and not greater than 3: ${}_5C_2$

*Those 2 slots must be 2: $1$

*Choose 2 slots (out of the remaining 3) to be greater than 3 and odd: ${}_3C_2$

*Those 2 slots must be 5: $1\times 1$

*The 1 remaining slot must be odd and not greater than 3, that is 1 or 3: $2$


Multiply the numbers within each case together and sum up 3 cases, then divide by $6^5$ to get $215/1944$.
