HOW TO FIND THE RANK OF A WORD IN DICTIONARY WITHOUT REPETITION

How to Find the Rank of a Word in Dictionary Without Repetition ?

For some students "Rank finding in dictionary problems" is a difficult topic in quantitative aptitude. Actually it is not like that. Once we understand the concept, it is not a difficult one to understand. 

How to Find the Rank of a Word in Dictionary Without Repetition - Practice question

Question 1 :

If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the ranks of the words (i) GARDEN (ii) DANGER.

Solution :

(i) GARDEN

Let us write the given word with alphabetical order,

A, D, E, G, N, R

Number of words starting with the letter A by rearranging the letters of the given word "GARDEN"

A _ _ _ _ _  =  5!

Number of words starting with the letter D

D _ _ _ _ _  =  5!

Number of words starting with the letter E

E _ _ _ _ _  =  5!

Number of words starting with the letter GAD

GAD _ _ _  =  3!

Number of words starting with the letter GAE

GAE _ _ _  =  3!

Number of words starting with the letter GAN

GAN _ _ _  =  3!

Number of words starting with the letter GAR

GAR _ _ _  =  1

The remaining letters are in alphabetical order.

3(5!) + 3(3!) + 1  =  3(120) + 3(6) + 1

  =  360 + 18 + 1

  =  379

Hence the total number of ways are 379.

(ii) DANGER

Let us write the given word with alphabetical order,

A, D, E, G, N, R

Number of words starting with the letter A by rearranging the letters of the given word "GARDEN"

A _ _ _ _ _  =  5!

Number of words starting with the letter DAE

D A E _ _ _  =  3!

Number of words starting with the letter DAG

D A G _ _ _  =  3!

Number of words starting with the letter DANE

D A N E _ _  =  2!

Number of words starting with the letter DANG

D A N G _ _  =  1

If we use the remaining letters in the alphabetical order.

5! + 2(3!) + 2! + 1  =  120 + 2(6) + 2 + 1

  =  120 + 12 + 3

  =  135

Hence the total number of ways are 135.

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