In the game of Yahtzee, you are rolling five fair dice, each of which can land on one of six sides. To compute the probability of observing exactly the numbers 1, 2, 3, 4, and 5 in any order (with no duplicates), we need to look at how many favorable outcomes fit this condition out of the total possible outcomes.

Step-by-step Solution:

1. Total possible outcomes:
   Since each die has 6 possible outcomes, rolling five dice simultaneously results in $6^5$ total possible outcomes: $6^5 = 7776$ So, there are 7776 possible outcomes when rolling five dice.

2. Favorable outcomes:
   The specific numbers we want (1, 2, 3, 4, and 5) must each appear exactly once. This means we need to arrange these five distinct numbers across the five dice in any order. The number of ways to arrange 5 distinct numbers in 5 positions is simply the number of permutations of 5 objects, which is: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

3. Probability of observing 1, 2, 3, 4, 5:
   The probability is given by the ratio of favorable outcomes to total possible outcomes: $\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{120}{7776} = \frac{5}{324}$

Thus, the probability of observing the numbers 1, 2, 3, 4, and 5 in any order when rolling five dice is $\frac{5}{324}$.

Would you like more details on this? Here are some related questions you could explore:

1. What is the probability of rolling exactly two sixes in a single roll of five dice?
2. How does the probability change if the dice are not fair?
3. What is the probability of rolling all five dice with the same number (a Yahtzee)?
4. How many possible ways are there to roll a full house (three of one number and two of another)?
5. What is the probability of rolling two pairs and one different number?

Tip: When dealing with dice or any probability questions, breaking the problem into total outcomes and favorable outcomes makes it easier to compute complex probabilities.