Question

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is (A) 360 (B) 192 (C) 96 (D) 48

Short answer

192

Step-by-step solution

Calculate Total Permutations

First, find the total number of ways the letters in COCHIN can be permuted. As there are 6 unique letters, the number of permutations is 6! (6 factorial).

Arrange Letters Alphabetically

List the given letters of the word COCHIN in alphabetical order: C, C, H, I, N, O.

Counting Permutations before 'COCHIN'

We calculate the number of permutations that come before the word COCHIN by fixing letters one by one from the beginning and permuting the rest of the letters.

Fixing Letter 'C'

The first letter 'C' is fixed because it is the smallest. We move on to the second letter.

Fixing Second Letter and Permuting Rest

The second letter can be 'C', 'H', 'I', 'N', or 'O'. For each letter before 'O', we find the number of permutations of the remaining 4 letters (5! / 2! because of the repeating 'C').

Calculate Permutations before 'O'

Before 'O', we have the letters 'C', 'H', 'I', 'N' which can be the second letter. We multiply the number of these letters by the permutations calculated in Step 5.

Add up Permutations

Sum up all the permutations calculated in Step 6 to find the total number of permutations before the word COCHIN.

Key concepts

Factorial

The concept of a factorial is fundamental in the field of combinatorics and permutations. A factorial, denoted as a number followed by an exclamation mark (e.g., 5!), represents the product of all positive integers from 1 up to that number. For instance, the factorial of 6 (6!) is calculated as:
\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\]
Factorials grow very rapidly with each additional number, which is why they are so crucial in calculating the number of permutations. It represents the total number of unique ways an entire set can be arranged. However, it's important to note that factorial of zero is defined as 1 (0! = 1).
In relation to our textbook exercise on the permutations of the word 'COCHIN', we use factorial to determine that there are 6! possible arrangements of these six letters, assuming they are all unique.

Permutation Formula

The permutation formula is used to calculate the number of possible arrangements of a set of items. When the items are distinct, the formula is simple, as it is just the factorial of the number of items. However, when there are repeating items, the permutation formula becomes more complex and is given as:
\[\frac{n!}{n_1! \times n_2! \times ... \times n_k!}\]
where \(n\) is the total number of items, and \(n_1, n_2, ..., n_k\) are the frequencies of each of the repeating items.
For example, in the word 'COCHIN', we have two 'C's. If we want to find the number of permutations with both 'C's considered indistinct, we would use the formula \(\frac{6!}{2!}\), with 6! being the total number of permutations of 6 items and 2! being the factorial of the frequency of 'C'.
Understanding this principle helps further explain the exercise where after fixing the first letter 'C' and choosing another letter before 'O', permutations of four remaining letters are calculated with the adjusted formula due to the repeating 'C'.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It has applications in many areas of mathematics, including algebra, geometry, and optimization, as well as in computer science and cryptography.
In our context, combinatorics helps us understand the different ways we can arrange the letters in a word like 'COCHIN' and how to count the arrangements following specific rules, like alphabetical order. The problem from the textbook is an application of combinatorial principles, where we are not just interested in counting the total permutations, but in finding the number of permutations that meet a certain condition, specifically those that come before a particular arrangement.
In this scenario, we apply systematic counting principles by fixing letters one by one and counting the permutations of the remaining letters, which is a common combinatorial approach for solving these types of problems. The exercise showcases how combinatorics is not just about how many, but also about the sequence of choices that lead to a particular arrangement or result.