Arranging Letters: Methods and Examples

Arranging letters in a word where some letters repeat can be a complex but interesting problem in combinatorics. This article explores the methods to calculate the number of ways to arrange the letters in the word rumour.

The Word 'rumour'

The word 'rumour' is a 6-letter word with the following distribution: 3 vowels (u, u, o) and 3 consonants (r, m, r).

The Basic Formula for Arrangement

The general formula to calculate the number of ways to arrange a word with repeated letters is given by:

Number of arrangements n! / (n?! × n?! × ... × n?!)

Where:

n is the total number of letters in the word. n?, n?, ..., n? are the frequencies of the repeated letters.

For the word 'rumour', we have:

Total number of letters, n 6 Letter R appears 2 times Letter U appears 2 times Letter M appears 1 time Letter O appears 1 time

Plugging these values into the formula, we get:

Number of arrangements 6! / (2! × 2! × 1! × 1!)

Step-by-Step Calculation

Let's break down the calculation:

6! 720 2! 2 2! × 2! × 1! × 1! 4 Number of arrangements 720 / 4 180

Therefore, the word 'rumour' can be arranged in 180 different ways.

The Word 'rumour' with Consonants Together

Another interesting problem is to find the number of ways to arrange the letters in 'rumour' such that the consonants (r, m, r) are always together. This can be treated as a single entity or 'block,' reducing the problem to arranging 4 entities (u, ○, o, u).

Step-by-Step Solution

Step 1: Treat the consonants (r, m, r) as one 'block' or entity. We now have 4 entities to arrange: u, ○, o, u. Step 2: Calculate the number of ways to arrange the 4 entities (u, ○, o, u). 4! / 2! 12 Step 3: Calculate the number of ways to arrange the 3 consonants (r, m, r) within the 'block.' 3! / 2! 3 Step 4: Multiply the number of arrangements of the entities by the number of arrangements of the consonants within the 'block.' Total number of ways 12 × 3 36

Therefore, there are 36 ways to arrange the letters in the word 'rumour' such that the consonants (r, m, r) are together.

Conclusion

This article has demonstrated two different methods to solve the problem of word arrangements with repeated letters. The first method involves calculating the total number of arrangements of all letters, while the second method treats certain letters as a single entity, simplifying the problem.

These techniques are fundamental in combinatorics and can be applied to a wide range of similar problems.