Question

If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in a dictionary, then the word SACHIN appears at serial number
A. 601
B. 600
C. 603
D. 602

Answer 1

Hint: To solve this question, we have to write the letters of the word SACHIN in the alphabetical order which leads to A, C, H, I, N, S. Let us consider the letter S coming in the first position. In the dictionary, words starting with S come only after all the words starting with the letters A, C, H, I, N are completed. We have to calculate the number of words that are starting with A, C, H, I, N to get to the words that are starting with S. Let us consider n characters to be arranged in r places. The number of ways is given by ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$.
The number of words starting with A is given by ${}^{5}{{P}_{5}}=$5! = 120 as there are 5 spaces left for C, H, I, N, S. Likewise we can calculate until the start of first letter S. Fixing S in first place, we go for the second position with 5 letters A, C, H, I, N and do the same process until we get the word SACHIN to get the serial number of SACHIN.

Complete step-by-step answer:
We have to arrange the letters of the word SACHIN in an alphabetical order. That is A, C, H, I, N, S. We have to consider the first position and calculate the number of words that come before the start of S in the list of all words.
Let us consider n characters to be arranged in r places. The number of ways is given by ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
For example, calculating the number of words that start with A, we get
C, H, I, N, S should be arranged in second to sixth positions (5 positions) of the word. That is n = 5 and r = 5. Using the formula, we get
${}^{5}{{P}_{5}}=5!$.
Similarly the words starting with C, H, I, N are 120 each.
The number of words before the words that start with S is = $5\times 120=600$.
Now we have fixed the first letter as S and go for the second letter. The word SACHIN has a second letter as A and we know that A is the first letter in the alphabetical order, so we can infer that the words with the first two letters SA comes after the 600 words. Likewise, C, H, I, N also in the alphabetical order and the word SACHIN comes after the 600 words. So, the serial number of the word SACHIN is 601.

So, the correct answer is “Option A”.

Note: In this type of serial number questions, the options are close enough to each other. A small mistake in the interpretation and calculation leads to a wrong answer by students. If we observe, there is no repeating letter in the word and if there are repeating letters, the process will be slightly different from the above process.