Permutation: Rank of word FATE

rockysheedy · 2007-01-14 06:29:25

Hello,

Here is a permutation question to find the rank of the word FATE:

Question:- If the letters of the word 'FATE' are arranged as in a dictionary, what is the rank of the word?

Solution:- The alphabetical order of the letter is A, E, F, T. The number of arrangements begining with A is got by putting A in the first place and arranging the other 3 letters. This can be done in 3 factorial ways.

Therefore the Number of arrangements which begin with A = 3 Factorial = 6
Therefore the Number of arrangements which begin with E = 3 Factorial = 6

The next arrangement is FAET = 1
The next arrangement is FATE = 1 --------------------- --------------- Why not the number of arrangement which begins with F is 3 factorial? Why only two arrangements starting with F are considered? There can be even other arrangement of the word FATE starting from F... like FTAE, FTEA, FEAT, FETA........why even these arrangements are not taken

Last edited by rockysheedy (2007-01-14 06:30:40)

mathsyperson · 2007-01-14 06:57:19

You're perfectly correct that those arrangements of FATE exist, but they're not considered because they're not needed to answer the question. You only needed to go through all the combinations until you found the FATE arrangement, because that will allow you to determine its rank.

There are 6 combinations that start with A, 6 that start with E, and then there's FAET and then FATE. This means that FATE is ranked 14th.

rockysheedy · 2007-01-15 00:02:12

Thank you for your reply.

I would like to know how exactly the placing takes place. Correct me if I am wrong.

Now the word 'FATE' has four letters F, A, T, E..........

We have to approach as per alphabetical manner like A, E, F, T.

Then for A there are 6 permutations and for E there are 6 permutation.

Now coming to F:- If F is the first letter of of the word there are remaining 3 places for remaining 3 letters that is for A, E, T now even these letters should be taken in alphabetical order.......like.....A, E and then T. So we get first two positions occupied by F and A and hence the word is.........FA

Now there are remaining 2 places for remaining 2 letters that is for E, T again take alphabet which is comes first for e.g. E comes before T hence take E. Now the arrangement becomes....FAE

Now place the last letter T in the last place and so the word becomes....... FAET-----------1

Now we have found out one permutation starting from F

To find another arrangements starting from letter F and second postion occupied by A so there are remaining 2nd positions for remaining 2 letters E and T. We have already found out the arrangement by fixing E in the 3rd position hence this time we place T in the 3rd position and place E in the 4th position The word that we get is.............FATE----------------------2

Now we have reached to the main word arrangement that is FATE hence we stop here as we were asked to find the rank of the word FATE otherwise we could have proceeded further by fixing E in the 2nd position and then putting A in the 3rd position and then T in the 4th position.

Then again fixing E in the 2nd position and interchanging the postion of A and T....

Then again fixing T in the first position and so on............... If we had to find out the rank of the word which would have started from T in this case.

I apologize for prompting. Does this comes under Permutation(Arrangements) or Combinations because you have mentioned about the combination. I might be wrong.

regards,
rocky sheedy.

Last edited by rockysheedy (2007-01-15 00:10:17)

mathsyperson · 2007-01-15 01:19:16

Yes, that method looks perfect.
And you're right in that it's a permutation question as well, as opposed to a combination one. I used combination in a casual sense rather than a mathematical one. Sorry for any confusino caused there.

Math Is Fun Forum

#1 2007-01-14 06:29:25

Permutation: Rank of word FATE

#2 2007-01-14 06:57:19

Re: Permutation: Rank of word FATE

#3 2007-01-15 00:02:12

Re: Permutation: Rank of word FATE

#4 2007-01-15 01:19:16

Re: Permutation: Rank of word FATE

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