Problem 157 - A Complete Resource Book in Mathematics for JEE Main

Question

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

Step-by-Step Solution

Answer

The word 'SACHIN' appears at serial number 601.

Step 1: Understanding the problem

We need to find the position of the word 'SACHIN' if all its permutations are arranged alphabetically.

Step 2: Identify permutations starting with letters before 'S'

First, determine how many permutations can be formed starting with letters that appear before 'S' in the alphabet (i.e., 'A', 'C', 'H', 'I', and 'N').

Step 3: Permutations starting with 'A'

There are 5 remaining letters: 'C', 'H', 'I', 'N', and 'S'. Calculate the permutations: \(5! = 120\).

Step 4: Permutations starting with 'C'

Similarly, permutations of 'A', 'H', 'I', 'N', 'S' are counted as: \(5! = 120\).

Step 5: Permutations starting with 'H'

Permutations of 'A', 'C', 'I', 'N', 'S' are: \(5! = 120\).

Step 6: Permutations starting with 'I'

Permutations of 'A', 'C', 'H', 'N', 'S' are: \(5! = 120\).

Step 7: Permutations starting with 'N'

Permutations of 'A', 'C', 'H', 'I', 'S' are: \(5! = 120\).

Step 8: Calculating permutations for words starting with 'S'

Now calculate how many permutations start with 'SA', 'SC', 'SH', and so on, until 'SACHIN' is reached.

Step 9: Words starting with 'SA'

For 'SA', the permutations of 'C', 'H', 'I', 'N' are \(4! = 24\).

Step 10: Words starting with 'SAC'

For 'Sac', permutations of 'H', 'I', 'N' are \(3! = 6\).

Step 11: Words starting with 'SACH'

For 'Sach', permutations of 'I', 'N' are \(2! = 2\).

Step 12: Words starting with 'SACHI'

For 'SACHI', permutation is 'N'. Only 1 such permutation exists.

Step 13: Position of the word 'SACHIN'

Add all the previous permutations: 5*120 + 24 + 6 + 2 + 1 + 1 = 601.

Key Concepts

Dictionary order of permutations

Permutations are all the possible arrangements of a set of items. In this case, the items are the letters of the word "SACHIN." When we talk about the "dictionary order of permutations," we mean arranging all permutations (or combinations) of the letters in the same order as they would appear in a dictionary.

Think of it like looking up words in a dictionary: words starting with 'A' come before those starting with 'B', and so forth. Following this order ensures a systematic method to find where any word would land if all permutations were listed. This is precisely what is required in the exercise: determining exactly where "SACHIN" would appear if you wrote out all its letter rearrangements alphabetically.

• It starts by considering permutations that begin with letters that come before 'S'.
• After accounting for all those, it moves to permutations starting with 'S', and narrows down to "SACHIN" specifically.

By adding these steps together, you find that "SACHIN" appears at position 601 in the list.

Factorial calculation

Permutations rely heavily on factorial calculations, denoted by the symbol "!" in mathematics. The factorial of a number, say 5!, is the product of all positive integers from 1 to that number. So, 5! equals 5 x 4 x 3 x 2 x 1 = 120.

In this exercise, factorial calculations help determine how many permutations are possible with the remaining letters after choosing the first letter. For instance, when considering permutations starting with 'A', there are 5 remaining letters to permute: 'C', 'H', 'I', 'N', and 'S'. Thus, 5! = 120 possible arrangements can be made for each letter choice that precedes 'S'.

• 5 letters -> 5! = 120 permutations
• 4 letters -> 4! = 24 permutations
• 3 letters -> 3! = 6 permutations

Understanding factorial calculations is fundamental to solving many permutations and combinations problems like the one above.

Alphabetical arrangement

Alphabetical arrangement, also known as lexicographical order, is simply arranging words based on the position of each letter in the alphabet. This arrangement helps us to sort and find words systematically, just like how words are sorted in dictionaries.

In solving the problem of finding "SACHIN"'s position among its permutations, each step starts with checking permutations that begin with letters appearing before 'S' alphabetically—that includes 'A', 'C', 'H', 'I', and 'N'.

• 'A' appears before 'C', so permutations starting with 'A' come first.
• Then, permutations starting with 'C', and so on, until reaching 'S'.

By considering these alphabetized sequences, and appropriately applying permutations with factorials, one can derive at which exact number the word "SACHIN" appears. This approach is a powerful tool not just in math, but in any scenario that requires ordering, such as coding and data organization.