Problem 61 - Precalculus with Limits

Question

Write all possible selections of two letters that can be formed from the letters \( A \), \( B \), \( C \), \( D \), \( E \), and \( F \). (The order of the two letters is not important.)

Step-by-Step Solution

Answer

The possible selections of two letters are: \(AB\), \(AC\), \(AD\), \(AE\), \(AF\), \(BC\), \(BD\), \(BE\), \(BF\), \(CD\), \(CE\), \(CF\), \(DE\), \(DF\), \(EF\). There are a total of 15 combinations.

Step 1 Calculate Total Number of Combinations

Here, there are a total of 6 letters, so \( n = 6 \), and the task is to find combinations of 2 letters, so \( r = 2 \). Using the combination formula: \( ^nC_r = \frac{n!}{r!(n-r)!} \), compute \( ^6C_2 = \frac{6!}{2!(6-2)!} \), which equals 15.

Step 2 Visualize All Possible Combinations

After calculating the total number of combinations, it's helpful to visualize all possible combinations of size 2 that can be made from the given 6 letters. The combinations are: \(AB\), \(AC\), \(AD\), \(AE\), \(AF\), \(BC\), \(BD\), \(BE\), \(BF\), \(CD\), \(CE\), \(CF\), \(DE\), \(DF\), \(EF\).

Key Concepts

Permutations

Permutations refer to the arrangement of objects where the order does matter. Imagine you have a set of objects and you want to know how many different ways you can arrange them. That's when permutations come into play. For instance, if you have three letters, say A, B, and C, the permutations include ABC, ACB, BAC, BCA, CAB, and CBA. Here, each arrangement is considered unique because the order changes. The formula to calculate the number of permutations of selecting and arranging \( r \) objects from a total of \( n \) is given by: \[ ^nP_r = \frac{n!}{(n-r)!} \]Where \( n! \) (n factorial) means multiplying all whole numbers from \( n \) down to 1. Unlike combinations, permutations take into account the different sequences available when arranging the objects. Permutations are instrumental when the arrangement order is critical for a task or situation.

Permutation vs Combination

Understanding the difference between permutations and combinations is key in combinatorics. While permutations consider the order of selection, combinations do not. This distinction is crucial depending on the scenario you face.

• Permutations: Are used when the arrangement order is important. Every different order is a unique permutation.
• Combinations: Are used when the arrangement order is not important. Two arrangements with the same items but different order are considered the same combination.

For example, if you are handed six letters and asked to select two, with permutations you'd count both AB and BA as two different options. However, in combinations, AB and BA are considered the same since the order doesn't matter. This understanding helps in problems where you need to decide whether the sequence of selection or arrangement of items matters or not. Utilizing the right concept greatly simplifies calculations in various scenarios.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arranging, and grouping objects. It's an umbrella term that encompasses both permutations and combinations among other concepts. The primary goal is to enumerate the number of possible arrangements and selections in a given scenario. This field finds applications in numerous areas like probability, statistics, computer science, and more. In our earlier task of selecting two letters from six, combinatorics helps determine the total number of possible selections without considering the order.By using the combination formula: \[ ^nC_r = \frac{n!}{r!(n-r)!} \]We can calculate precisely how many ways we can choose \( r \) objects from \( n \). In the example provided, we found that there are 15 ways to select two letters from six, which showcases the core utility of combinatorics in simplifying selection problems without order concern.The practical use of combinatorics extends beyond mathematics, aiding in efficient decision-making and problem-solving strategies in real-world situations.