Rank of a Word in Dictionary

A common type of problem asked in many examinations is to find the 'rank' of a given word in a dictionary. What this means is that you are supposed to find the position of that word when all permutations of the word are written in alphabetical order. we use the concept of the rank of words to find the position of the word in the dictionary.

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This Story also Contains

1. Factorial notation
2. The rank of a word - without repetition of letters
3. The rank of a word - with repetition of letters
4. Solved Examples
5. Example 1: All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is : [JEE MAINS 2023]

In this article, we will cover the Rank Of A Word In the Dictionary. This topic falls under the broader category of Permutations and combinations, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE.

Factorial notation

Many times we multiplied consecutive integers. On the basis of this factorial notation is devised. In the counting principle many times answer is written in the form of factorial to save us time. The product of first n natural numbers is denoted by n! and is read as 'factorial n'.

The rank of a word - without repetition of letters

Example: Find the rank of a word MATHS in a dictionary made using its letters

Step 1: Write down the letters in alphabetical order.

The order will be A, H, M, S, T.

Step 2: Find the number of words that start with a superior letter.

Any word starting from A will be above MATHS. So, if we fix A at the first position, we have 4! = 24 words. (number of ways arranging H, M, S, T).

Similarly, there will be 24 words that will start with H.

The number of words starting with MAH is 2! = 2

The number of words starting with MAS is 2! = 2

The number of words starting with MATHS is 1! = 1

Therefore, the overall rank of the word MATHS is 24 + 24 + 2 + 2 + 1 = 53

The rank of a word - with repetition of letters

Example: Find the rank of the word INDIA in a dictionary made using its letters

Write down the letters in alphabetical order, the order will be A, D, I, I, N.

• The number of words starting with A is 4!/2! = 12 (We are dividing by 2! because I is repeating itself)
• The number of words starting with D is 4!/2! = 12
• The number of words starting with IA is 3! = 6 (number of ways arranging I, D, N)
• The number of words starting with ID is 3! = 6
• The number of words starting with II is 3! = 6
• The number of words starting with INA is 2! = 2
• The number of words starting with INDA is 1! = 1
• The number of words starting with INDIA is 1! = 1

Therefore, the overall rank of the word INDIA is 12 + 12 + 6 + 6 + 6 + 2 + 1 + 1= 46

Summary

The concept of the rank of a word helps in solving combinatorial puzzles and algorithmic challenges. Understanding the rank of a word provides a methodical approach to determining its position when all possible permutations of its letters are arranged alphabetically. It enhances one's ability to analyze and manipulate permutations effectively.

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Solved Examples

Example 1: All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is : [JEE MAINS 2023]

Solution

$\begin{aligned} & \mathrm{B}=5!=120 \\ & \mathrm{C}=5!=12 \\ & \mathrm{I}=5!=120 \\ & \mathrm{~L}=5!=120\end{aligned}$
$
\begin{aligned}
& \mathrm{PB}=4!=24 \\
& \mathrm{PC}=4!=24 \\
& \mathrm{PI}=2!=24 \\
& \mathrm{PL}=2!=2
\end{aligned}
$
$\begin{aligned} & \text { PUBC }=2!=2 \\ & \text { PUBI }=2!=2 \\ & \text { PUBLC }=1 \\ & \text { PUBLIC }=1\end{aligned}$

Rank =582

Hence, the answer is 582

Example 2: The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in the English dictionary. Then the serial number of the word 'MANKIND' is [JEE MAINS 2022]

Solution

$\left(\frac{4 \times 6!}{2!}\right)+(5!\times 0)+\left(\frac{4!\times 3}{2!}\right)+(3!\times 2)+(2!\times 1)+(1!\times 1)+(0!\times 0)+1=$

Hence, the answer is '1492'.

Example 3: If the letter of the word 'MOTHER' is permuted and all the words so formed (with or without meaning) are listed as in a dictionary, then the position of the word 'MOTHER' is [JEE MAINS 2021]

Solution

MOTHER : E,H,M,O,R,T

We need to find the position of the word MOTHER in the dictionary.

Starting with E: 5!

Starting with H: 5!

Starting with ME: 4!

Starting with MH: 4!

Starting with MOE: 3!

Starting with MOH: 3!

Starting with MOR: 3!

Starting with MOTE: 2!

The next word is MOTHER: 1

Total:= 309

Hence, the answer is 309

Example 4: If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in the dictionary, then the word SACHIN appears at serial number:

Solution: Rank of SACHIN
The alphabetical order is A, C, H, I, N, S
Number of words that start with $A \Rightarrow 5!$
Number of words that start with $C \Rightarrow 5!$
Number of words that start with $H \Rightarrow 5!$
Number of words that start with $I \Rightarrow 5!$
Number of words that start with $N \Rightarrow 5!$
Next word is $S A C H I N \Rightarrow 1$
Position of $\mathrm{SACHIN}=5(5!)+1=601$

Hence, the answer is 601

Example 5: If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in the English dictionary, then the position of the word QUEEN is:

Solution: calculate the rank of QUEEN.
The alphabetical order of letters is E, E, N, Q, U
Word starting with $E \rightarrow 4!=24$
Word starting with $N \rightarrow \frac{4!}{2!}=12$ (As 2 E's are there)

Word starting with $Q E \rightarrow 3!=6$
Word starting with $Q N \rightarrow \frac{3!}{2!}=3$
Word starting with $Q U E E N \rightarrow 1$
So, the position of QUEEN $=24+12+6+3+1=46^{\text {th }}$

Hence, the answer is 46