Problem 40 - Algebra and Trigonometry

Question

These problems involve distinguishable permutations. Arranging Letters In how many different ways can the letters of the word ELEEMOSYNARY be arranged?

Step-by-Step Solution

Answer

There are 39,916,800 possible arrangements of the letters in 'ELEEMOSYNARY'.

Step 1: Count the Total Letters

First, count the total number of letters in the word 'ELEEMOSYNARY'. There are 12 letters in total.

Step 2: Identify Repeated Letters

Identify any letters that are repeated in the word. The letter 'E' appears 3 times, and the letter 'Y' appears 2 times.

Step 3: Apply the Permutation Formula

Use the formula for distinguishable permutations: \[ \frac{n!}{a! \cdot b! \cdot c! \cdots} \] where \( n \) is the total number of letters, and \( a, b, c, \ldots \) are the frequencies of the repeated letters. Here, \( n = 12 \), \( a = 3 \) for the letter 'E', and \( b = 2 \) for the letter 'Y'.

Step 4: Write Out the Formula

Plug the values into the formula: \[ \frac{12!}{3! \times 2!} \].

Step 5: Calculate Factorials

Calculate the factorials: \( 12! = 479,001,600 \), \( 3! = 6 \), and \( 2! = 2 \).

Step 6: Substitute and Solve

Substitute back into the formula: \[ \frac{479,001,600}{6 \times 2} = \frac{479,001,600}{12} = 39,916,800 \].

Key Concepts

Permutation Formula

When dealing with permutations, especially when arranging letters like in the word 'ELEEMOSYNARY', we need to count how many unique ways we can arrange these letters. The permutation formula for distinguishable permutations is ideal here. This formula is given by:

• \( \frac{n!}{a! \cdot b! \cdot c! \cdots} \)

Where:

• \( n \) is the total number of letters.
• \( a, b, c, \ldots \) are the counts of each repeated letter.

This formula helps us by correcting for the over-counting of arrangements caused by repeated letters. It ensures every letter arrangement is unique by dividing out the permutations of the repeated letters.

Factorials

To effectively use the permutation formula, understanding factorials is crucial. A factorial, denoted as \( n! \), is a product of all positive integers less than or equal to \( n \). For example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 2! = 2 \times 1 = 2 \)

Factorials rapidly increase in value as \( n \) grows, as seen with \( 12! = 479,001,600 \). In permutations with repeated elements, factorial calculations are used in both the numerator (for the total permutations of all items) and the denominator (to divide out permutations of repeated items).
Factorials simplify the process of finding total unique arrangements by reducing larger numbers into a product of smaller sequential numbers.

Repeated Letters

Repeated letters in a word can affect how we calculate permutations. For 'ELEEMOSYNARY', the letter 'E' is repeated 3 times, and 'Y' is repeated twice. This repetition means that simply using the factorial of the total letter count would overestimate the number of unique permutations.
When letters repeat, we have several identical items that can switch places without resulting in a new permutation. That's why we divide by the factorials of the counts of these repeated letters (\(3!\) for 'E' and \(2!\) for 'Y') to adjust our calculations and avoid over-counting.

• Identify all repeated letters.
• Calculate their factorials.
• Use these in the permutation formula's denominator.

Arranging Letters

Arranging letters to find unique permutations can seem complex at first, especially with larger words. The objective is to determine how many distinct ways the letters can be ordered. Using 'ELEEMOSYNARY', with its mix of unique and repeated letters, showcases this well.
Start by counting all letters — here, it's 12. Recognize patterns such as repeated letters. Then apply the permutation formula to calculate the unique arrangements:

• Count the total letters: 12
• Identify repeated letters: 'E' appears 3 times, 'Y' appears 2 times
• Insert these into the formula \( \frac{12!}{3! \times 2!} \) for simplification

The result tells you exactly how many different sequences or orders the word letters can be arranged, making it both an enjoyable and enlightening mathematical exploration.