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All 5! = 120 permutations of the word SMART are listed.  What is the position of the word SMART?

 Jun 14, 2020
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OK, so the word "SMART" starts with S. We first want to know the approximate position of words that start with "S".

 

Arranging the letters in alphabetical order, we get "AMRST". That means words start with "S" is on a comparatively low position.

 

In reality, if we fix the first letter, there are 24 ways to arrange the other letters. So there are 24 words which start with "A", 24 which start with "M", so on and so on.

 

At the time we reach "S" finally, there would have been at least 72 words in front of the current word.

 

So the list of permutation looks like this:

AMRST

AMRTS

...

...

 

SAMRT

SAMTR

...

...

 

STRMA

...

...

 

So we need to know where does words which starts with "SM" lies.

 

When we fix the first two letters, there are 6 ways to arrange the other three.

 

So there are 72 + 6 words before the first word which starts with "SM".

 

And we also know that the first word which starts with "SM" is SMART, because "ART" is in alphabetical order.

 

Therefore, the position of the word is 72 + 6 + 1 = 79

 

Edited: Sorry, I did basic arithmetic wrongly at the very last step.

 Jun 14, 2020
edited by MaxWong  Jun 14, 2020

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