I think it is a simple one..

1. probability for the first person to the button for one of the floors would be - 6/6
since he would be having six possible choice out of 6 choices (like if we toss a coin with both the faces as heads, the probability of getting of a head is 2/2)
2. as the question is asking for exactly each floor, the probability for second person to press the other floor than the first one would be 5/6
3. similarly for third person would be 4/6
4. 3/6
5. 2/6
6. 1/6

ans would be (6/6 * 5/6 8 4/6 * 3/6 * 2/6 * 1/6) = 6!/6^6

"A"

1st person has 6 options for leave at a floor
2nd ...........5 ............................
.
.
.
6th ...........1 ............................
Therefore, Total no. of ways for 6 person to leave at 6 different floor= 6!

Total no of ways in which 6 people can leave without any restrictions. Each one has 6 options
Therefore Total no of ways without any restrictions = 6*6*6*6*6*6

Hence Prob.= Total no. of ways for 6 person to leave at 6 different floor/Total no of ways without any restrictions = 6!/6*6*6*6*6*6 = Option A