Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Find the number of distinguishable permutations of the letters in the word moon. List all the permutations.

Step-by-Step Solution

Answer

There are 12 distinguishable permutations of the word "moon".

Step 1: Calculate Total Letters and Their Frequencies

First, identify the total number of letters in the word "moon". The word has 4 letters in total: 'm', 'o', 'o', and 'n'. Here, two letters 'o' are repeated.

Step 2: Use Permutation Formula for Repeated Elements

To find the number of distinguishable permutations of the letters, use the formula for permutations with repeated elements: \[\frac{n!}{p_1! \times p_2! \times \cdots \times p_k!}\] where \(n\) is the total number of letters, and \(p_1, p_2, \ldots, p_k\) are the frequencies of the repeated elements. Here, \(n = 4\), \(p_1 = 2\) for letter 'o'. Substitute these into the formula: \[\frac{4!}{2!} = \frac{24}{2} = 12\]. Thus, there are 12 distinguishable permutations.

Step 3: List All Distinguishable Permutations

List each permutation making sure each collection of letters forms a unique arrangement: 1. moon 2. moNo 3. mOon 4. mONo 5. monO 6. moNO 7. oomn 8. oonm 9. onom 10. onMo 11. omNo 12. Omno

Key Concepts

Permutations with Repeated Elements

When it comes to arranging letters, the concept of permutations with repeated elements can simplify calculations. In a set where items repeat, some permutations would look identical, making them not unique. Hence, the formula to calculate the number of distinguishable permutations takes into account these repetitions: \[\frac{n!}{p_1! \times p_2! \times \cdots \times p_k!}\] - Here, \(n\) represents the total number of items to arrange.- \(p_1, p_2, \ldots, p_k\) are the counts of each repeated element.By dividing the total number of permutations by the factorial of these counts, we remove the duplicate arrangements.For example, in the word "moon," the letter 'o' repeats twice, leading to fewer unique permutations than if all letters were distinct.

Factorial

The factorial is a fundamental concept in permutations and combinations, representing the product of an integer and all the integers below it. It is denoted by the symbol '!', and its basic definition is:\[ n! = n \times (n - 1) \times (n - 2) \times \ldots \times 2 \times 1 \]- **Example**: \(4! = 4 \times 3 \times 2 \times 1 = 24\)Factorial is pivotal in calculating permutations because it tells us how many different ways we can arrange a set number of items. The process becomes streamlined when considering repeated elements, as seen in permutations with repeated elements. By using factorials, we easily adjust the total count by accounting for duplicates, making the math straightforward and manageable.

Letter Arrangement

Arranging letters can be both fun and complex, especially when repetitions are involved. In combinatorics, letter arrangements refer to all unique ways to position letters of a word to form meaningful or non-meaningful combinations. To calculate how many distinct ways you can organize a word like "moon," each permutation must be considered. - **Unique vs. Non-unique Arrangements**: - Unique arrangements account for repeats, e.g., "moon" has fewer unique permutations due to the repeated 'o'. - Non-unique refers to treating each letter as different, which would result in more arrangements. By methodically applying the formula for permutations with repeated elements, we ensure no arrangement is overlooked or counted more than once. This careful approach allows us to explore all possible formations without redundancy.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations, essential for solving problems like arranging letters in a word. It provides systematic methods for counting the number of ways objects can be arranged or selected from a group. In combinatorics, permutations are sequences where order matters. Here, distinguishable permutations require considering identical items to avoid overcounting. This is where our permutation formula for repeated elements fits perfectly. - **Why Combinatorics Matters**: - It helps organize and simplify complex counting problems. - Useful in probability, computer science, and decision-making processes. Understanding combinatorics enriches our problem-solving toolkit, offering an organized way to tackle diverse counting challenges with clarity and precision.