Problem 31 - College Algebra

Question

For the following exercises, find the distinct number of arrangements. The letters in the word "academia"

Step-by-Step Solution

Answer

There are 6720 distinct arrangements of the letters in "academia."

Step 1: Count Total Letters

First, we need to determine the total number of letters in the word "academia." The word has 8 letters.

Step 2: Identify Repeated Letters

Identify letters that repeat in the word. In "academia", the letter 'a' appears 3 times, and the letter 'c' appears 1 time, with the remaining letters appearing only once.

Step 3: Apply Permutation Formula for Repeated Elements

To find the number of distinct arrangements, use the permutation formula for repeated elements: \[ \frac{n!}{p_1! \times p_2! \times \, ... \, \times p_k!} \] where \( n \) is the total number of letters, and \( p_1, p_2, ..., p_k \) are the frequencies of the repeated letters. For "academia", this becomes: \[ \frac{8!}{3!} \] since 'a' repeats 3 times.

Step 4: Calculate Factorials

Calculate the factorials: \( 8! \) is 40320 and \( 3! \) is 6.

Step 5: Compute the Distinct Arrangements

Substitute these factorials into the formula: \[ \frac{8!}{3!} = \frac{40320}{6} = 6720 \] Thus, there are 6720 distinct arrangements of the letters in "academia".

Key Concepts

Factorials in Permutations

Factorials play a critical role when determining permutations. But what exactly is a factorial? In mathematics, the factorial of a number is the product of all positive integers up to that number.
For example, when we write the factorial of 8 as \(8!\), it means:

• Calculate: \(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320\)

We use factorials in permutations to account for the arrangement orders of a set.
In problems with repeated elements, we adjust the standard permutation calculation by dividing by the factorials of the count of each repeated element, as shown in the formula:

• \[ \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

Here, the numerator \(n!\) represents all possible arrangements of the letters, while each \(p_i!\) in the denominator removes the duplicate arrangements caused by the repetition of letters.
Thus, without factorials, finding permutations involving repetitions would be impossible to calculate correctly.

Distinct Letter Arrangements

When analyzing distinct letter arrangements, we determine how to organize characters uniquely without repeating the exact sequence.
This concept can appear challenging at first because of repeated characters in a word, like 'a' in "academia," which appears three times.
To correctly find the number of distinct arrangements, one needs to consider:

• Total number of letters
• Frequency of each repeated letter

By using the formula for permutations of multi-set letters

• \[ \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

we can calculate how these repeats impact the total arrangements. Here, \(n\) is the total letters, and each \(p_i\) is the repetition of a specific letter.
For "academia," this calculation eliminates the duplicate arrangements caused by the three 'a's, resulting in 6720 unique permutations without disregarding any possible orderings of the letters.

Combinatorics Formula

Combinatorics, a branch of mathematics focusing on counting, is where we develop formulas to tackle arrangement problems like the one seen in this exercise.
The combinatorics formula used here, \(\frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}\), lets us effectively manage problems involving permutations with repeated elements.
Key elements of this formula include:

• Determining the total number of items (\(n\)) you're arranging
• Identifying repetitions of items to find each \(p_i\)

This formula helps balance the equation by subtracting the identical permutations that stem from repeated items.
It is straightforward but incredibly powerful, giving us the precise count of unique permutations, which can otherwise be difficult to handle manually for more extensive and intricate problems. By leveraging this combinatorics technique, students effectively learn to apply mathematical reasoning to real-world situations requiring logic and precision.