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October 17th, 2018, 04:33 AM   #1
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Words counting and ranking

Hi everyone,

Can you support me to solve this problem;

All words which contain 2,3,4, or 5 English letters (A to Z) are listed alphabetically. In each word, the letters can be repeated, but any two adjacent letters must be distinct. So, the first word is "AB", and the last one is "ZYZYZ".

* What is the number of words in this list?

* What is the rank of the word "PAPER"?

* What is the 234th word?

Can anyone help me, please?
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October 19th, 2018, 03:30 AM   #2
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Any help please?
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October 19th, 2018, 06:38 AM   #3
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Quote:
Originally Posted by Hussain2629 View Post
Any help please?
NO.
Thanks from Joppy
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October 19th, 2018, 07:35 AM   #4
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There are 26×25 = 650 two-letter words, the 234th being JI.

There are 26×25×25 = 16250 three-letter words.

etc.
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October 20th, 2018, 02:16 AM   #5
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Quote:
Originally Posted by skipjack View Post
there are 26×25 = 650 two-letter words, the 234th being ji.

There are 26×25×25 = 16250 three-letter words.

Etc.
Thank you skipjack,

I found out the number of words in this way;

$\displaystyle 26 × 25^1 + 26 × 25^2 + 26 × 25^3 + 26 × 25^4 = 10579400
$ words.

But how did you find the 234th word? And how to find the rank of the word "PAPER"?
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October 20th, 2018, 05:32 AM   #6
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Consider how many words precede JA and how many precede PABAB.
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October 20th, 2018, 10:39 PM   #7
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Quote:
Originally Posted by skipjack View Post
Consider how many words precede JA and how many precede PABAB.
Dear skipjack,

The number of words before "JA" is much greater than 234, because if we consider the words which start with "A" only;

*that have two letters* = 1 x 25 = 25
*that have three letters* = 1 x 25 x 25 = 625
*that have four letters* = 1 x 25 x 25 x 25 = 15625
*that have five letters* = 1 x 25 x 25 x 25 x 25 = 390625

So, there are (25+625+15625+390625) or 406900 words start with "A", all of them are before "JI". Therefore, the 234th word must start with "A".
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October 21st, 2018, 07:22 AM   #8
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You're correct. The 234th is ABAJX. The 6112737th is PAPER.
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October 29th, 2018, 11:38 AM   #9
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5 days of trying, but could not find the right way
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