Question

The letters of the word ‘COCHIN’ are permuted and all the permutations are arranged in alphabetical order as in English dictionary. The number of words that appear before the word ‘COCHIN’ is
(a) 360
(b) 192
(c) 96
(d) 48

Answer 1

Hint: We solve this problem by arranging the letters in the word COCHIN in the ascending order of letters.
Then we use all the permutations that lead to the word COCHIN by fixing the letters one by one.
We have the condition that the number of ways of arranging \[n\] objects is given as \[n!\]
We have the other condition that the number of ways of arranging \[n\] objects in which \[r\] are identical is given as \[\dfrac{n!}{r!}\]

Complete step by step answer:
We are given with the word COCHIN
Let us arrange each letter in the order of alphabets then we get
C, C, H, I, N, O

Here, we can see that there are six letters
Let us assume that there are six boxes in which the letters need to be arranged as follows

Now, let us place the first letter C in the first box then we get

Here, we can see that the first letter of COCHIN and the first letter of our arrangement is the same.
So we can say that the first box is fixed with letter C
Now, let us place the second letter C in the second box then we get

Here, we can see that the second letter is not matching with COCHIN
So, let us arrange the remaining 4 letters in four other boxes.
We know that the condition that number of ways of arranging \[n\] objects is given as \[n!\]
By using the above condition we get number of ways of arranging 4 letters is \[4!\]
Now, let us place the next letter that is H in the second box then we get

Here, we can see that the second letter is not matching with COCHIN
So, let us arrange the remaining 4 letters in four other boxes.
By using the condition of the arrangement we get number of ways of arranging 4 letters is \[4!\]
Here, we can see that the total number of arrangements until now as
\[\begin{align}
  & \Rightarrow 4!+4!=24+24 \\
 & \Rightarrow 4!+4!=48 \\
\end{align}\]
Now, let us place the next letter that is I in the second box then we get

Here, we can see that the second letter is not matching with COCHIN
So, let us arrange the remaining 4 letters in the other four boxes.
By using the condition of the arrangement we get the number of ways of arranging 4 letters is \[4!\]
Here, we can see that the total number of arrangements until now as
\[\Rightarrow 48+4!=48+24=72\]
Now, let us place the next letter that is N in the second box then we get

Here, we can see that the second letter is not matching with COCHIN
So, let us arrange the remaining 4 letters in the other four boxes.
By using the condition of the arrangement we get the number of ways of arranging 4 letters is \[4!\]
Here, we can see that the total number of arrangements until now as
\[\Rightarrow 72+4!=72+24=96\]
Now, let us place the next letter that is O in the second box then we get

Here, we can see that the second letter in COCHIN is matching with our permutations
So, we can fix the two letters in the first two boxes
Now, let us place the first letter from the remaining letters that is C in the third box then we get

Here, we can see that the third letter in COCHIN is matching with our permutations
So, we can fix the three letters in the first three boxes
Now, let us place the first letter from the remaining letters that is H, I, and N the remaining boxes respectively then we get

Here, we can see that there are a total of 96 words before the word COCHIN
So, option (c) is the correct answer.

Note: Students may make mistakes in taking the number of words before the given word and the position of the given word. If we are asked to find the position of COCHIN then we need to add 1 to 96 to get the position because the arrangement of letters to get COCHIN is also a permutation. But we are a number of words that are before COCHIN then there is no need to add 1 to 96 to get the correct answer.