Question

The letters of the word COCHIN are permuted and all permutations are arranged in alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
(a) 96
(b) 48
(c) 183
(d) 267

Answer 1

Hint: First of all, arrange the letters of the words COCHIN in alphabetical order. Now, we know that before COCHIN, words starting from CC, CH, CI, and CN would appear. So, count all the words and their sum would be the required answer.

Complete step by step answer:

Here, we are given that the word COCHIN is permuted and all the permutations are arranged in alphabetical order as in the English dictionary. We have to find the number of words that appear before the word COCHIN. Before proceeding with the question, let us first talk about a few terms.
1. Permutation: In Mathematics, permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging its elements, a process called permuting.
2. Combination: The combination is a way of selecting items from a collection. Unlike permutations, the order of selection does not matter in combination.
Now, let us consider our question. Let us first arrange the letters of the word COCHIN in alphabetical order like in the dictionary
C C H I N O
We can see that in the dictionary, the first would start with CC, that is
\[\underline{C}\text{ }\underline{C}\text{ }\underline{4}\times \underline{3}\times \underline{2}\times \underline{1}\]
Now, we have the remaining 4 places and four letters to fill. So, we get the total words starting from
\[CC=4\times 3\times 2\times 1=4!=24\]
Now, after CC, the next set of words would start with CH, that is
\[\underline{C}\text{ }\underline{H}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}\]
Now, again we have four places and four letters left to fill. So, we get total words starting from
CH = 4! = 24
After CH, the next set of words would start with CI, that is,
\[\underline{C}\text{ }\underline{I}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}\]
Now, again we have four places and four letters left to fill. So, we get total words starting from
CI = 4! = 24
After CI, the next set of words would start with CN, that is
\[\underline{C}\text{ }\underline{N}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}\text{ }\underline{{}}\]
Now, again we have four places and four letters left to fill. So, we get total words starting from
CN = 4! = 24
After CN, the next set of words would start with CO, that is our required set of words because COCHIN starts with CO.
In all the words starting from CO, the first words would start with COC because C would come before any letter, and as we can see that COCHIN also starts with COC, so COCHIN would be the first word in this list.
Hence, we get the total words that appear before COCHIN = Total words starting from CC, CH, CI, and CN.
= 24 + 24 + 24 + 24
= 96 words
Therefore, option (a) is the right answer.

Note: In questions involving the arrangement as in a dictionary, students must analyze each set of words properly because students often miss one or another set of words while carrying the other. Also, each letter of each word must come according to the alphabetical order. It is better to arrange each letter one by one and then take their sum. Also, there is no special formula for repetition in the case of these questions.