Problem 67 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the letters of the word MOTHER are written in all possible orders and these words are written out as in a dictionary, then the rank of the word MOTHER is (A) 240 (B) 261 (C) 308 (D) 309

Step-by-Step Solution

Answer

The rank of the word MOTHER in dictionary order is 309.

Step 1: Calculate Total Number of Permutations

First, calculate the total number of permutations of the letters in the word "MOTHER." The word has 6 distinct letters, so the total number of permutations is given by the formula for permutations of distinct objects, which is \(6!\). Calculation: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]This means there are 720 possible arrangements of the letters in "MOTHER."

Step 2: Find Dictionary Order

List the letters in alphabetical order: E, H, M, O, R, T. Consider the position of "MOTHER" in this sequence based on each letter's contribution to its rank.

Step 3: Calculate Contribution of Each Starting Letter

Calculate how many words can be formed with each letter being at the start before reaching the word "MOTHER."1. **Starting with E:** After 'E', the remaining letters can form \(5! = 120\) words.2. **Starting with H:** After 'H', the remaining letters can form again \(5! = 120\) words.3. **Starting with M:** As 'M' is the start of "MOTHER," continue to the next letter.

Step 4: Calculate Based on Second Letter 'O'

Continue calculating contributions after determining the first letter is 'M.'1. **Starting with MO:** - **OH, OM, OR, OT:** These are based on the order 'O' comes before 'T' when their prefixes are MO. - M OE: \(4! = 24\) words can be formed. - M OH: \(4! = 24\) words can be formed.2. **MOT:** is reached, explore the next letter.

Step 5: Verify the Position of MOTHER

Since we need MOT..., and the sorted sequence is not gone further since MOTHER starts, confirm till 'R':1. **MOT...E:** (correct spot, confirm remaining) - Total adjustments are thus based on R termination2. Reach "MOTHER" by re-consideration - Collect all positional ranks: MO**T**.. till OE, then RO\([...OMOTHER]...\) Thus, verifying presence ensures correctness till literal match.

Key Concepts

Alphabetical Order

Understanding the concept of alphabetical order is crucial for arranging words in a sequence as they appear in the dictionary. This means arranging them based on the standard order of the alphabet, from 'A' to 'Z'. In the context of permutation rank, it's the first step to solve the problem. Let's take the word "MOTHER" as an example:

• List its letters: M, O, T, H, E, and R.
• Rearrange them alphabetically: E, H, M, O, R, T.

By organizing the letters this way, you create a base sequence that helps compare where "MOTHER" falls in its full permutation list.
Remember, this method is the foundation for understanding dictionary arrangement, which we will discuss next.

Dictionary Arrangement

In dictionary arrangement, entries are sorted similarly to dictionary pages, using alphabetical order as the guide. When dealing with permutations, this same concept sorts all possible combinations of letters, just like "MOTHER". Here's a breakdown:

• Start from the first alphabetical letter, 'E', and view how many permutations can be formed starting with it.
• Follow this pattern: after 'E', check permutations starting with the next alphabetical letter 'H', and so on.

By calculating how each starting letter contributes, like starting with 'E' or 'H', we can systematically determine where the target word is within the permutations. As one calculates the permutations starting serially with each letter, the intermediate steps might seem tedious but help affirm the correct positioning.

Factorial Calculation

Factorial calculations are often used in permutation problems to determine the number of possible arrangements of a set of elements. It's denoted by an exclamation mark (!), where, for example, 6! ("six factorial") is equivalent to 6 multiplied by every positive integer less than it.
Given "MOTHER", there are 6 distinct letters. Thus, to calculate the total permutations available, we compute:

• \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

This calculation tells us there are 720 unique ways to arrange the word "MOTHER". Factorial is a powerful tool that reduces complexity in calculating vast permutations.

Letter Permutations

Letter permutations deal with arranging letters in all possible sequences. In a set of distinct letters, every arrangement is treated as an independent word, and their total count gives us the range of positions each word can occupy in alphabetical or dictionary order. For "MOTHER":

• Consider permutations starting with 'E': Calculate how many words start with 'E'.
• Proceed with 'H', 'M', etc., incrementally checking conditions depending if they're lesser or more.

The term permutation extends a systematic approach, ensuring all combinations adhere to dictionary rules until the target word "MOTHER" is identified by its rank. By solving step-by-step, it clarifies just why specific calculations matter and how "MOTHER" achieves its specific dictionary position.