Question

A domino is made of two squares, each of which is marked with one, two, three, four. five, or six spots or is left blank. A set of dominoes consists of dominoes with all possible pairs showing in the two squares. How many different dominoes are there in a set?

Short answer

There are 28 different dominoes in a set.

Step-by-step solution

Understanding the Problem

A domino is composed of two squares. Each square can display between 0 and 6 spots. We need to calculate the total number of unique dominoes that can be formed using all combinations of these numbers on the two squares.

Calculate Total Combinations

Each square can have one of 7 different values (0-6). Thus, there are a total of \(7 \times 7 = 49\) possible combinations for the numbers that can appear on the two squares without considering any restrictions.

Eliminate Duplicates

In dominoes, order doesn't matter (e.g., 2-3 is the same as 3-2), so we need to count only unique pairs. This can be calculated using combinations: \(\binom{7+1}{2} = \binom{8}{2} = 28\).

Calculate the Combination

The mathematical formula for combinations is \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). For our problem: \(\binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28\).

Verify the Solution

Verify that the combination count covers all unique arrangements and considers mirrored pairs as identical. The calculation of 28 ensures that we're counting only unique pairs.

Key concepts

Permutations

Permutations play a crucial role in combinatorics when we deal with the arrangement of items where the order matters. However, in the context of our domino exercise, permutations are not directly applicable because the order on each pair of dominoes does not matter.

When considering dominoes like "2-3" and "3-2," order is irrelevant; therefore, you would consider these the same. That's why in solving this problem, combinations are used instead of permutations. Permutations would be helpful if you needed to calculate sequences where the position impacts meaning, which can be crucial in understanding concepts of arrangement and probability involving different scenarios. Despite not using permutations in this example, knowing the difference is valuable. Understanding permutations is essential for problems involving ordered lists or rankings, such as arranging books on a shelf or creating passwords where the sequence is important. It is always important to identify when the order of selection is significant before diving into calculations.

Combinations

Combinations are all about selecting items where order doesn't matter. This is precisely what we addressed in our original domino problem.

In the exercise, each domino pair is unique regardless of the order of the numbers (like "2-3" being identical to "3-2"). To find unique dominoes, we use combinations rather than permutations.

The mathematical formula to calculate combinations is given by:
- \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Here, \( n \) represents the total items to choose from, and \( r \) is the number of items to select. In your domino problem, there are 7 possible values (0 through 6) and you effectively select 2 at a time, accounting for pairs without regard to order, which equals 28 unique dominoes using \( \binom{8}{2} \). This calculation covers all pairs and ensures each pair is only counted once.

Understanding how to calculate combinations is useful for many scenarios where the arrangement is irrelevant, such as forming teams, choosing colours, or selecting items without concern for their order in daily life or mathematical contexts.

Discrete Mathematics

Discrete mathematics forms the backbone of our analysis regarding dominoes and combinations. This field of study focuses on countable, distinct, and often finite structures. The principles of discrete math are foundational for problems involving permutations and combinations, just like in our exercise with dominoes.

Discrete mathematics covers various topics, including: -

• Graph theory, which deals with networks and connections.
• Combinatorics, the art of counting and arranging sets.
• Logic, to understand propositions and their validity.
• Number theory, which relates to integers and their properties.

Understanding discrete mathematics enables problem solvers to handle various real-world applications, including computer science's algorithms and programming logic.

The domino problem is an excellent example of discrete mathematics in action. By using its concepts, we can determine the number of unique domino combinations without repetition efficiently. This discipline allows us to manage finite structures logically and effectively, crucial for understanding and solving complex problems in various fields.