Problem 11 - Algebra and Trigonometry

Question

Find the number of distinguishable permutations of the given letters. $$ A A A B B C $$

Step-by-Step Solution

Answer

There are 60 distinguishable permutations.

Step 1: Identify the total number of items

First, count the total number of letters in the set. We have 6 letters: three A's, two B's, and one C. Thus the total number of items is 6.

Step 2: Calculate the factorial for total items

To find the total number of ways to arrange 6 items, we calculate the factorial of 6. This is written as 6! and calculated as follows: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

Step 3: Calculate factorials for repeated items

Since some letters are repeated, we need to divide by the factorials of the number of each repeated letter to account for indistinguishable permutations.- For the letter A, which occurs 3 times, calculate 3!:\[ 3! = 3 \times 2 \times 1 = 6 \] - For the letter B, which occurs 2 times, calculate 2!:\[ 2! = 2 \times 1 = 2 \]

Step 4: Compute the permutations considering repetitions

Use the formula for distinguishable permutations, which is: \[ \frac{n!}{n_1! \times n_2! \times n_3! \ldots} \]where \( n \) is the total number of items, and \( n_1, n_2, \ldots \) are the counts of each repeated letter.Substitute the values:\[\frac{6!}{3! \times 2!} = \frac{720}{6 \times 2} = \frac{720}{12} = 60\]

Step 5: Conclude the solution

Having performed the calculations, the number of distinguishable permutations of the letters AAA, BB, and C is 60.

Key Concepts

Factorials

Factorials are a mathematical way to describe the product of a sequence of descending natural numbers. When you see the symbol "!", it denotes a factorial. For example, 6!, pronounced "six factorial,"means \( 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \).

Factorials are incredibly helpful for calculating the total number of ways to arrange (permute) a set of objects. Each object's position in a permutation matters, and factorials simplify these calculations by systematically accounting for all possible arrangements.

• 5! (Five factorial) = \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• 4! (Four factorial) = \( 4 \times 3 \times 2 \times 1 = 24 \)

Factorials can grow very large very quickly as numbers increase. Understanding how to calculate and work with them is fundamental in combinatorial mathematics and various fields involving probability and statistics.

Indistinguishable Permutations

When dealing with permutations, you might encounter situations where some items are identical or indistinguishable from one another. This is important, because simply calculating arrangements using factorials as though all items were different would overestimate the true number of unique arrangements.

To find the number of distinguishable permutations of a set with indistinguishable items, you use the formula: \\[ \frac{n!}{n_1! \times n_2! \times \ldots} \] \where:

• \( n \) is the total number of items.
• \( n_1, n_2, \ldots \) are the counts of each type of indistinguishable item.

In our example, with the letters AAA, BB, and C,even though there are 6 letters, many permutations are indistinguishable due to repeated characters. So, instead of calculating 6!, we divide by the permutations of the repeated letters 3! for A's and 2! for B's. This ensures we only count unique permutations. Understanding indistinguishable permutations helps correctly calculate possibilities whenever duplicate items are involved.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns in sets. It deals with problems of how to count objects while considering various constraints, such as identical items or specific orderings.

There are many aspects of combinatorics, but some essential concepts include:

• Permutations: Arrangements of objects where the order matters.
• Combinations: Selections of objects where the order does not matter.
• Factorials: Used for calculating permutations and combinations.

In the realm of combinatorics, factorials are often used to determine permutations. Meanwhile, combinations use a different formula — not involving factorials of the different items — because order doesn’t matter in combinations.

Applications of Combinatorics

Combinatorics is widely used in fields like computer science, operations research, and statistical analysis for practical problem-solving. Identifying the number of possible arrangements (like seating charts or lineup possibilities) and combinations (like lottery numbers or grouping strategies) can be solved using these principles. By grasping these concepts, you'll be better prepared to tackle diverse mathematical and real-world challenges involving arrangement and selection of items.