Question

How many ways can one choose one right glove and one left glove from six pairs of different gloves without obtaining a pair?

Short answer

There are 30 ways to choose one right glove and one left glove without getting a matching pair.

Step-by-step solution

Understand the Problem

We need to calculate the number of ways to choose one right glove and one left glove such that they do not form a matching pair. Each of the 6 pairs has distinct gloves.

Count Total Combinations

First, determine how many total combinations of one right glove and one left glove are possible. Since there are 6 right gloves and 6 left gloves, there are 6 x 6 = 36 total combinations.

Count Matching Pairs

There are 6 matching pairs (one for each pair of gloves). These must be excluded from the total combinations to avoid selecting a pair.

Calculate Non-Pair Combinations

Subtract the number of matching pairs from the total combinations to get the number of non-pair combinations: 36 total combinations - 6 matching pairs = 30 non-pair combinations.

Key concepts

Distinct Pairs

When we talk about distinct pairs in the context of combination problems, we refer to a pair of items where each item is unique in its own attribute. In the glove selection problem, each pair of gloves comprises two distinct entities: a right glove and a left glove. Every pair is considered distinct because they are unique, meaning their features or properties are different from one another. This distinctiveness is crucial when calculating combinations because it ensures that each selection is unique. Without this uniqueness, each combination could inadvertently repeat, thus skewing the calculations. Remember, the key in distinct pairs is maintaining the uniqueness among all combinations.

Non-Matching Combinations

In combinatorics, the idea of non-matching combinations is all about selecting items in such a way that no two selected items match in predefined criteria. In our glove selection problem, this means choosing a right glove and a left glove that do not belong to the same pair. To solve this, we first calculate all possible combinations of right and left gloves, which is simply multiplying their counts together. For instance, if we have 6 right gloves and 6 left gloves, we multiply them to get 36 possible combinations. However, among these, 6 are matching pairs where both gloves are from the same set. To find the non-matching combinations, which are our focus, we subtract these matching pairs from the total combinations. This leaves us with the desired combinations that do not repeat the same pair.

Glove Selection Problem

The glove selection problem is a delightful example of how basic combinatorial principles can be applied to solve real-life puzzles. In problems like these, one must carefully consider the conditions imposed on the selection process, such as choosing from non-matching pairs. The challenge lies in understanding how to navigate through choices logically. For our specific case with 6 pairs of gloves, the task is to select a right and left glove from different pairs, ensuring they don’t form a complete set. Such questions bolster your understanding of how restrictions play a role in combinations. By eliminating choices that meet certain criteria (like matching pairs), you naturally arrive at the correct count of combinations that fit the problem’s conditions. It’s a great exercise in logical deduction and counting, showcasing the practical applications of combinatorics.