Well let's think.
We have a die of 6 sides.. but the sides are labeled 1,2,3,4,5,5.
We're tossing the die 5 times and trying to figure out the probability of getting 1,2,3,4,5 in any order.
To start off, the first number we roll can be either 1,2,3,4, or 5 to satisfy what we're trying to do.
What are the chances we roll 1,2,3,4, or 5 the first time? Well the die only has sides 1,2,3,4,5 and 5, so you will HAVE to get either 1,2,3,4 or 5 the first time you roll. So, the probability for the first roll is 6/6 or 1.
Now, the first time we rolled, we either rolled a 1,2,3,4 or 5. So, since we want to roll one of each number, whatever we rolled the first time, we cannot roll again. For example-sake, let's say that the first roll, we rolled a 1. We cannot roll a 1 again, because then we will have two 1s, but we need to roll one of each number.
So, now we need to roll either a 2,3,4, or 5. Our chances of rolling one of these are 5/6.
Let's say for simplicity that we rolled a 2 on the second roll. Now we have a 1 and a 2. We need either a 3,4, or 5. What are the chances of rolling one of these? 4/6.
Let's say, again for simplicity, we rolled a 3 on the third roll. Now we have a 1,2, and 3. We need a 4 or 5. What are the chances of rolling one of these? 3/6.
Now we have, let's say, 1,2,3, and 4. We just need the 5. Since there are two sides on the die labeled 5, our chances of rolling a 5, is 2/6.
Now we just multiply all the probabilities together.
We end up with or .