Key Concepts and Formulas

1. Probability Definition: The probability of an event $E$ is the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space. $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
2. Fundamental Principle of Counting (Multiplication Principle): If an event can occur in $m$ ways and a second independent event can occur in $n$ ways, then the two events can occur in $m \times n$ ways. This principle extends to any number of independent events.
3. Permutations: A permutation is an arrangement of objects where the order of arrangement matters. The number of permutations of selecting $r$ distinct items from a set of $n$ distinct items is given by: ${}^nP_r = \frac{n!}{(n-r)!}$

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Step-by-Step Solution

Step 1: Determine the Total Number of Possible Outcomes (The Sample Space)

• Understanding the Experiment: We are rolling three standard six-sided dice. Each die has 6 possible outcomes (1, 2, 3, 4, 5, or 6). Since the outcome of one die does not affect the others, these are independent events.
• Applying the Multiplication Principle:
  • The first die can show any of its 6 faces.
  • The second die can show any of its 6 faces.
  • The third die can show any of its 6 faces.
• Calculation: The total number of unique sequences of outcomes for the three dice is the product of the possibilities for each die. Total number of outcomes $= 6 \times 6 \times 6 = 6^3 = 216$.

Step 2: Determine the Number of Favorable Outcomes (Getting Different Numbers on the Three Dice)

• Understanding the Favorable Event: We need to find the number of outcomes where all three dice show distinct numbers. This means the number on the first die must be different from the second, and both must be different from the third.
• Why Order Matters (Permutations): When we roll three dice, we consider them distinguishable (e.g., first die, second die, third die). Therefore, rolling (1, 2, 3) is different from rolling (2, 1, 3). This implies that the order in which the different numbers appear on the dice is important, which is a characteristic of permutations.
• Applying Permutations (Method 1: Direct Counting with Constraints):
  • For the first die, there are 6 possible numbers it can show.
  • For the second die, since its number must be different from the first, there are 5 remaining choices.
  • For the third die, its number must be different from both the first and second dice, leaving 4 remaining choices.
  • Using the multiplication principle for these constrained choices: Number of favorable outcomes $= 6 \times 5 \times 4 = 120$.
• Applying Permutations (Method 2: Using the Permutation Formula):
  • We are selecting 3 distinct numbers from the 6 available faces (1, 2, 3, 4, 5, 6) and arranging them among the 3 dice.
  • Here, $n=6$ (total distinct numbers available) and $r=3$ (number of dice/positions to fill).
  • Using the permutation formula: ${}^6P_3 = \frac{6!}{(6-3)!} = \frac{6!}{3!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} = 6 \times 5 \times 4 = 120.$
• Conclusion: The number of favorable outcomes is 120.

Step 3: Calculate the Probability

• Applying the Probability Formula: $\text{Required Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ $\text{Required Probability} = \frac{120}{216}$
• Simplifying the Fraction: We need to simplify the fraction to its lowest terms, where the numerator $p$ and denominator $q$ are co-prime.
  • We can divide both the numerator and denominator by their greatest common divisor (GCD).
  • $120 = 2^3 \times 3 \times 5$
  • $216 = 2^3 \times 3^3$
  • The GCD of 120 and 216 is $2^3 \times 3 = 8 \times 3 = 24$.
  • $\frac{120 \div 24}{216 \div 24} = \frac{5}{9}$
• Result: The probability of getting different numbers on the three dice is $\frac{5}{9}$.

Step 4: Determine $p$ and $q$ and Calculate $q-p$

• Matching the Given Form: The problem states the probability is $\frac{p}{q}$, where $p$ and $q$ are co-prime. From our calculation, $\frac{p}{q} = \frac{5}{9}$.
• Verifying Co-prime Condition: The numbers 5 and 9 are co-prime because their only common positive divisor is 1.
• Identifying $p$ and $q$: We have $p=5$ and $q=9$.
• Calculating $q-p$: $q-p = 9 - 5 = 4$

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Common Mistakes & Tips

• Distinguishable vs. Indistinguishable Items: In most probability problems involving multiple dice, coins, or similar items, they are considered distinguishable unless explicitly stated otherwise. This means (1,2,3) is distinct from (2,1,3) for three dice. This is crucial for counting total and favorable outcomes.
• Permutations vs. Combinations: A common pitfall. Use permutations when the order of selection matters (as in this problem, where the order of different numbers on distinct dice matters). Use combinations when only the selection of items matters, and their order is irrelevant.
• Simplifying to Co-prime Terms: Always simplify the probability fraction to its lowest terms. The condition that $p$ and $q$ are co-prime ensures a unique pair of values for $p$ and $q$. Failing to simplify would lead to incorrect values for $p$ and $q$ and thus for $q-p$.

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Summary

This problem involves calculating the probability of a specific event when rolling multiple dice. The solution followed a structured approach: first, determining the total number of possible outcomes using the multiplication principle ($6^3 = 216$); second, identifying and counting the favorable outcomes where all three dice show different numbers, which requires permutations ($^6P_3 = 120$); third, calculating the probability by dividing favorable outcomes by total outcomes and simplifying the fraction to its co-prime form ($\frac{120}{216} = \frac{5}{9}$); and finally, using the identified values of $p=5$ and $q=9$ to compute $q-p=4$.

The final answer is $\boxed{4}$, which corresponds to option (B).