Problem 19 - 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 is 309.

Step 1: Total Number of Permutations

We start by calculating the total number of permutations of the letters in the word MOTHER. The word MOTHER has 6 different letters. The number of permutations, which is also the total number of different words that can be made by rearranging all letters, is given by the factorial of the number of letters: \(6! = 720\).

Step 2: Arrange Alphabetically

Next, we arrange the letters of the word MOTHER in alphabetical order to see the sequence in the dictionary: E, H, M, O, R, T.

Step 3: Find the Position of MOTHER

Identify where MOTHER fits, starting with the first letter and accounting for how many permutations can exist before it. 1. Start with M. Before M, there are no permutations starting with letters E or H that can come before MOTHER, as EHEMTR is the next block starting with H after M. 2. Keep M fixed, compare the second letter. After “M”, consider all possible letters that come before O in alphabetical order. - E: Permutations starting with ME: 5! = 120 - H: Permutations starting with MH: 5! = 120 Total permutations starting with ME and MH = 120 + 120 = 240 So, there are 240 permutations before reaching MO_. Start MO_. 3. Fix MO, continue with the third letter. The next positions of O are organized: - E: Permutations starting with MOE: 4! = 24 - H: Permutations starting with MOH: 4! = 24 Total permutations starting with all wages: 264 till MOH 4. MOH is written and continue with next letters - Fix MOH and consider all places permutation can happen, no letter inbetween. - MOHE_ has two possible paths so forth. One (MOT_), next is examined matching. Finally, the target MOTHER is reached at permutation 309 in all listing. Thus, the rank of MOTHER is 309.

Key Concepts

Factorial

Factorial is a mathematical operation that is essential in calculating permutations. It is denoted by an exclamation mark (!). Factorial of a number, say 6, is the product of all positive integers less than or equal to that number, which can be represented as:\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]Using factorial allows us to determine the total number of possible arrangements of a set of distinct items (like the letters in "MOTHER"). This is essential when setting the stage for finding permutations that help order words alphabetically, as done in dictionaries.

Alphabetical Order

Alphabetical order is the sequence that is commonly used in dictionaries, starting with the lowest alphabetical character and progressing to the highest. When working with permutations, arranging elements in alphabetical order allows us to compare them effectively. For example, in the word "MOTHER," the letters are M, O, T, H, E, R. We arrange them alphabetically as: E, H, M, O, R, T. This order helps identify all possible permutations leading up to the target permutation "MOTHER," ensuring accuracy when determining its rank relative to other combinations.

Dictionary Order

Dictionary order, sometimes called lexicographical order, is the method of sorting sequences (like strings or words) into an order that mirrors the way words are arranged in a dictionary. Given the alphabetical order E, H, M, O, R, T for "MOTHER," we start permutations beginning with each subsequent letter, ensuring all possibilities that could precede "MOTHER" are accounted for. This step-by-step way of arranging permutations as they would appear in a dictionary ensures each sequence can be methodically placed until reaching the desired word.

Rank of a Word

The rank of a word is essentially the position of a particular permutation (or word) when all permutations of its letters are listed in dictionary order. Calculating rank involves determining the number of permutations that fit before a given word within the alphabetical listings. Following the process for "MOTHER," fixing letters one by one and calculating permutations of the remaining letters allows us to determine how many permutations precede our word, ultimately leading us to its exact position or rank, such as 309 in this case. Understanding how to calculate a rank can help with various problems involving permutations and combinations.