Rank of a word in Dictionary

Rank of a word is the position of that word, when we arrange the words formed by alphabets of that given word in dictionary order. Lets see an example.

To understand this type of problem just assume that we have only 3 letters A, B and C in the English alphabets then the dictionary will have total meaning full or meaning less 3! = 6 words and their order is-

ABC → 1ST word

ACB→ 2nd Word

BAC → 3rd Word

BCA → 4th Word

CAB → 5th Word

CBA → 6th Word

So if I ask you what is the rank of word CAB the answer is 5th.

Example) If all the letters of the word MOTHER is arranged in dictionary format then find the rank/position of the word MOTHER.

Solution)

(i) Arrange all the alphabets of the word MOTHER in alphabetical order like (E, H, M, O, R, T)

Now strike off the first letter M from (E, H, M, O, R, T)

Then count the number of letters before M, and it is equal to 2, which is the coefficient of 5!.

Then strike off the next letter O (E, H, M, O, R, T)

Then count the number of letters before O and it is equal to 2 which is coefficient of 4!

Then strike off the next letter T (E, H, M, O, R, T)

Then count the number of letters before T and it is equal to 3 which is coefficient of 3!

Then strike off the next letter H (E, H, M, O, R, T)

Then count the number of letters before H and it is equal to 1 which is coefficient of 2!

Then strike off the next letter E (E, H, M, O, R, T)

Then count the number of letters before E and it is equal to 0 which is coefficient of 1!

Then the next word is MOTHER

So Rank of the word MOTHER is 2(5!) + 2(4!) +3(3!) + 1(2!) + 0(1!) + 1= 240 + 48+18+2+1 = 309

Hence rank of word MOTHER is 309.

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