Name: Permutations: Rank of a Word Without Repetition
Uploaded: 2019-07-19
Duration: 25 min 18 s
Description: If all the words formed by rearranging the letters of a given word are arranged as in a dictionary, how do we find the rank of a particular word among them?

Question

If the letters of the word AJANTA are permuted in all possible ways and the words thus formed are arranged in dictionary order.

Find the ranks of the words (i) AJANTA (ii) JANATA

Accepted Answer

The dictionary order of the letters of the word AJANTA is A A A J N T

In the dictionary order first comes that words which begin with the letter A.

If we fill the first place with A, we may set the word AJANTA.

Second place can be filled with A, the remaining $4$ places can be filled in $4! = 24$ ways.

On proceeding like this, we get

AA --- $= 4! = 24$

AJAA -- $= 2! = 02$

AJ ANA----- $= 1 = 0 1$

AJANTA $= 1 = 01$

Therefore, Rank of the word AJANTA

$= 24 + 02 + 01 + 01 = 28$

In the dictionary order first comes that words which begin with the letter A.

If we fill the first place with A, remaining $5$ letters can be arranged in $\frac{5!}{2!}$ ways (since there $2$ A's remain) On proceeding like this, we get

A ------ $= \frac{5!}{2!} = 60$

J A A- -- $= 3! = 6$

JANAA - - $= 1 = 1$

JANATA $= 1 = 1$

Therefore, the rank of the word JANATA is $= 60 + 6 + 1 + 1 = 68 .$

If the letters of the word AJANTA are permuted in all possible ways and the words thus formed are arranged in dictionary order.

Find the ranks of the words (i) AJANTA (ii) JANATA

Important Questions on Permutations and Combinations

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