Counting Arrangements of Repeating Letters: Examples

Table of Contents

1. Introduction
2. Permutations and Combinations
3. Arrangements of Letters
4. Example Word: Mississippi
5. Other Examples
6. How to Count Arrangements
7. How to Count Four-Letter Words
8. Example Word: Independence
9. FAQs

Arrangements of Letters with Repetition

Arranging letters with repetition can be a tricky task, especially when there are multiple letters that repeat. In this article, we will discuss how to calculate the number of ways in which such words can be arranged. We will take the example word "Mississippi" to illustrate this concept.

Before we dive into the example, let's revise the basics of permutations and combinations. Permutations refer to the number of ways in which a set of objects can be arranged in a specific order. Combinations, on the other HAND, refer to the number of ways in which a set of objects can be arranged without regard to order.

Example Word: Mississippi

The word "Mississippi" has 11 letters, with some letters repeating. To calculate the number of possible arrangements, we first need to take account of all the letters and how many times they repeat. The letter "M" appears only once, the letter "I" appears four times, the letter "S" appears four times, and the letter "P" appears twice.

To calculate the number of possible arrangements, we use the formula:

n! / (p! q! r! ...)

where n is the total number of items, and p, q, r, etc. are the number of items that are alike of one kind.

For the word "Mississippi," the number of possible arrangements is:

11! / (4! 4! 2!) = 34,650

Other Examples

Let's take another example to illustrate this concept. Suppose we have three letters: A, B, and C. There are three factorial (3!) ways to arrange these three letters: ABC, ACB, BAC, BCA, CAB, and CBA.

Now, let's repeat one of these letters. If we replace the letter "C" with the letter "A," we get the word "ABA." In this case, there are only three possible arrangements: AAB, ABA, and BAA. This is because the two "A"s cannot be interchanged, and so the number of possible arrangements is reduced.

In general, the number of permutations of n things taken all at a time, in which p things are alike of one kind, q things are alike of the Second kind, and so on, is:

n! / (p! q! r! ...)

How to Count Arrangements

To count the number of arrangements of a word with repeating letters, we first need to take account of all the letters and how many times they repeat. We then use the formula:

n! / (p! q! r! ...)

where n is the total number of items, and p, q, r, etc. are the number of items that are alike of one kind.

Let's take the example of the word "Mississippi" again. The word has 11 letters, with some letters repeating. To calculate the number of possible arrangements, we first need to take account of all the letters and how many times they repeat. The letter "M" appears only once, the letter "I" appears four times, the letter "S" appears four times, and the letter "P" appears twice.

Using the formula, we get:

11! / (4! 4! 2!) = 34,650

How to Count Four-Letter Words

To count the number of four-letter words that can be formed using the letters of a word with repeating letters, we first need to select four letters out of the given word. The four letters need not be all different; they could be various other possibilities. Once we have selected the four letters, we can determine the number of arrangements of those four letters.

Let's take the example of the word "Mississippi" again. To count the number of four-letter words that can be formed using the letters of this word, we first need to select four letters out of the 11 letters in the word. There are several cases to consider:

1. All four letters are different: There are four choices of letters (M, I, S, N, P), and we need to select all four. This can be done in 4C4 ways. The four letters can be arranged in 4! ways. Together, the selection and arrangement of these four letters can be done in 4! ways.

2. Two letters are alike and two are different: We have three choices for the two letters that are alike (II, SS, PP). Let's suppose we select II. We then need to select two different letters out of the remaining three letters (M, S, P). This can be done in 3C2 ways. The four letters can be arranged in 4!/2! ways (because two letters are alike). Together, the selection and arrangement of these four letters can be done in 3C1 x 3C2 x 4!/2! ways.

3. Two letters are alike and two others are alike: We have a choice of three letters that are repeating (II, SS, PP). Let's suppose we select SS. The four letters can be arranged in 4!/2!2! ways (because two letters are alike and two others are alike). Together, the selection and arrangement of these four letters can be done in 3C2 x 4!/2!2! ways.

4. Three letters are alike and one is different: We can select any one of the letters that repeat three times (II, SS). Let's suppose we select II. We then need to select the one different letter out of the remaining three letters (M, S, P). This can be done in 3C1 ways. The four letters can be arranged in 4!/3! ways (because three letters are alike). Together, the selection and arrangement of these four letters can be done in 3C1 x 3C1 x 4!/3! ways.

5. All four letters are alike: We can select any one of the letters that repeat four times (I, S). Let's suppose we select S. The four letters can be arranged in 4!/4! ways (because all four letters are alike). Together, the selection and arrangement of these four letters can be done in 2C1 x 4!/4! ways.

Adding up the number of possible four-letter words from all five cases, we get:

24 + 108 + 18 + 24 + 2 = 176

Example Word: Independence

Let's take another example word: "independence." The word has 12 letters, with some letters repeating. To calculate the number of possible arrangements, we first need to take account of all the letters and how many times they repeat. The letter "D" appears three times, the letter "E" appears four times, and all other letters appear only once.

Using the formula, we get:

12! / (3! 4!) = 27,720

FAQs

Q: What is the difference between permutations and combinations? A: Permutations refer to the number of ways in which a set of objects can be arranged in a specific order. Combinations refer to the number of ways in which a set of objects can be arranged without regard to order.

Q: How do You count the number of arrangements of a word with repeating letters? A: To count the number of arrangements of a word with repeating letters, you first need to take account of all the letters and how many times they repeat. You then use the formula: n! / (p! q! r! ...), where n is the total number of items, and p, q, r, etc. are the number of items that are alike of one kind.

Q: How do you count the number of four-letter words that can be formed using the letters of a word with repeating letters? A: To count the number of four-letter words that can be formed using the letters of a word with repeating letters, you first need to select four letters out of the given word. The four letters need not be all different; they could be various other possibilities. Once you have selected the four letters, you can determine the number of arrangements of those four letters.