Question

How many ways are there to roll 10 dice so that all six different faces show?

Short answer

There are 10,324,128 ways to roll 10 dice so that all faces 1 to 6 appear at least once.

Step-by-step solution

Understand the Problem

We need to calculate the number of ways to roll 10 dice so that each of the six different faces (1 through 6) appears at least once.

Apply the Inclusion-Exclusion Principle

Use the inclusion-exclusion principle to count the number of ways all faces appear. First, calculate the total number of unrestricted outcomes for rolling 10 dice, then subtract configurations where at least one face is missing.

Calculate Total Outcomes

The total number of outcomes when you roll 10 dice is since each die has 6 faces. Therefore, the total is \(6^{10}\).

Calculate Outcomes with At Least One Face Missing

Calculate the number of ways where at least one face is absent. Let \(S_i\) be the set with the \(i\)-th face missing. Using the inclusion-exclusion principle, calculate: \[ |S_i| = 5^{10} \] (one face missing) for 6 terms, then apply inclusion-exclusion for combinations of more than one face missing.

Calculate Each Inclusion-Exclusion Term

- Exclusion: Add the number of ways two faces are missing. \[ |S_{i,j}| = 4^{10} \] for \(C(6,2)\) terms.- Further Exclusion: Add for three faces missing \( |S_{i,j,k}| = 3^{10} \), continue for four, five, and six faces missing.

Compute Final Count via Inclusion-Exclusion Principle

Using the inclusion-exclusion principle, plug in the computed values:- Calculate the included outcome for single face absent, subtract when needed for additional absent faces.- Total desired outcome is: \[ 6^{10} - C(6,1) \, 5^{10} + C(6,2) \, 4^{10} - C(6,3) \, 3^{10} + C(6,4) \, 2^{10} - C(6,5) \, 1^{10} \] where \(C(n,k)\) are combinations.

Find Final Result

Perform arithmetic calculations to find the final number of ways using a calculator or computing software to evaluate large powers and combinations precisely.

Key concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a key concept in combinatorics that helps in counting the number of elements in the union of several sets. It avoids the problem of overcounting by correcting for overlaps between sets. To understand this principle, consider three sets: A, B, and C. The idea is to start with the sum of the sizes of each set, subtract the sizes of the intersections for each pair of sets, add back the size of the intersection of all three sets, and so forth. The formula for three sets is:

• \( |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C| \).

This principle is particularly useful in problems where constraints affect certain elements, requiring the careful exclusion and inclusion of outcomes.

Dice Probability

Understanding dice probability involves determining how likely certain outcomes are when rolling dice. Each die has 6 faces, numbered 1 through 6, making each face equally likely to appear when rolled. For a single die, the probability of any specific face showing is \( \frac{1}{6} \). When multiple dice are rolled, as in the problem of rolling 10 dice, probabilities require considering all combinations of outcomes. With 10 dice, the potential combinations are vast, and their calculations involve employing principles like the Inclusion-Exclusion Principle, especially when certain outcomes, such as all 6 faces appearing, are desired. Proper setup and breakdown of the problem into manageable parts are crucial in deriving the correct probability and understanding how constraints impact the total possible outcomes.

Combinatorial Counting

Combinatorial Counting is fundamental in determining the number of ways objects can be arranged or combined under certain conditions. Core techniques include using formulae for permutations and combinations, represented mathematically as \( P(n,k) \) and \( C(n,k) \), respectively. Permutations count arrangements where order matters, while combinations count group selections where order does not. For rolling dice, since each outcome of a die roll is independent, combinations can express the multiply layered arrangements possible, such as choosing subsets of dice to roll particular numbers. In the problem of rolling 10 dice to see all faces at least once, combinatorial counting assists in applying the Inclusion-Exclusion Principle, accounting for how dice configurations exclude or include certain face appearances.

Mathematical Problem Solving

Mathematical Problem Solving involves applying logical processes to solve complex problems. This includes understanding the problem, using appropriate mathematical strategies, and employing systematic trial and error combined with refined techniques like combinatorial methods. For problems involving probability and combinatorial counting, breaking down the task into steps is helpful. Identifying what is sought, calculating possible outcomes, recognizing constraints like excluding dice configurations, and checking work ensures accuracy. Employing tools such as algorithms, calculators, or software for high order mathematical tasks keeps computation feasible. This is especially important in exercises, such as the dice problem, where solving large-scale calculations like powers and combinations is necessary for arriving at the correct solution.