Problem 166 - A Complete Resource Book in Mathematics for JEE Main

Question

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is: (A) \(58^{\text {th }}\) (B) \(46^{\text {th }}\) (C) \(59^{\text {th }}\) (D) \(52^{\text {nd }}\)

Step-by-Step Solution

Answer

The position of the word SMALL is 52nd, which corresponds to option (D).

Step 1: Identify Distinct Letters

The word SMALL consists of the distinct letters S, M, A, and L. There are 5 letters in total with 'L' repeating.

Step 2: Calculate Total Permutations

Calculate the total number of permutations of the word using all available letters: \( \frac{5!}{2!} = 60 \), as 'L' repeats twice.

Step 3: Rank Based on Dictionary Order

List the letters in alphabetical order: A, L, M, S. The first letter for the formation of the word is determined as each letter appears.

Step 4: Compute Contribution of A

Words starting with A: \( \frac{4!}{2!} = 12 \). These are the permutations of remaining letters L, M, S, L.

Step 5: Compute Contribution of L

Words starting with L: \( \frac{4!}{2!} = 12 \). These permutations are not counted because no word starting with L can precede SMALL.

Step 6: Prefix 'M' Contributions

Words starting with M: \( \frac{4!}{2!} = 12 \) words are possible before reaching those starting with S.

Step 7: Prefix 'S','A' Contributions

When leading with S, consider second letter A: \( \frac{3!}{2!} = 3 \). These are permutations of letters L, L, M.

Step 8: Shortest Arrangement for SMALL

Consider 'S','M','A': 1st and then arrange L, L following. Placement is directly after SMAAL, SMALA and SMALL.

Step 9: Determine Position of 'SMALL'

Thus, SMALL is positioned immediately after 46 words (12A + 12L + 12M + 3 for SA + 6 for SL), which places it at position 52.

Key Concepts

Dictionary Order

When arranging words or letters, the dictionary order is just like how words appear in a dictionary – alphabetically. In this exercise, we are asked to find the position of the word "SMALL" assuming all possible permutations of its letters are arranged as they would be in a dictionary. Starting with the smallest letter and moving to the next, we create a sequence of words by fitting all the letters in every possible order. For instance, if we have the letters A, L, M, and S from "SMALL", the words starting with A come first when arranged alphabetically, followed by those starting with L, and so on. The goal is to determine where "SMALL" falls in this sequence.

Repeated Letters

Repeated letters can complicate permutations because not every letter contributes a unique result. Due to the repetition, some arrangements will look the same. In "SMALL", we have the letter 'L' occurring twice. As a result, to find unique permutations, we must divide the total number of permutations by the factorial of the number of repeated letters. Thus, for the word "SMALL" with 5 letters where 'L' repeats, the total permutations are computed by dividing by the factorial of the repetitions:

• Calculate: \( \frac{5!}{2!} \)
• The divisor \(2!\) accounts for the repeated 'L'.

This ensures that the count reflects unique orderings, excluding inaccurate duplicates that arise solely due to repeated elements.

Factorial

Factorial is a key concept in permutations and combinations, helping us determine how many different ways we can arrange things. Represented by the symbol \(!\), a factorial of a number is the product of all integers from 1 up to that number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorial is crucial when calculating the number of arrangements of a set of items. In our specific exercise, we divide by a factorial to adjust for repeated letters, specifically for the letter 'L' in `SMALL`. The factorials tell us how to account for all possible ways the letters can be uniquely placed into the slots provided.