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 October 17th, 2018, 03:33 AM #1 Member   Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1 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?
 October 19th, 2018, 02:30 AM #2 Member   Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1 Any help please?
October 19th, 2018, 05:38 AM   #3
Math Team

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From: Ottawa Ontario, Canada

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Quote:
 Originally Posted by Hussain2629 Any help please?
NO.

 October 19th, 2018, 06:35 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 There are 26×25 = 650 two-letter words, the 234th being JI. There are 26×25×25 = 16250 three-letter words. etc.
October 20th, 2018, 01:16 AM   #5
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From: Saudi Arabia

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Quote:
 Originally Posted by skipjack 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"?

 October 20th, 2018, 04:32 AM #6 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 Consider how many words precede JA and how many precede PABAB.
October 20th, 2018, 09:39 PM   #7
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From: Saudi Arabia

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Quote:
 Originally Posted by skipjack 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".

 October 21st, 2018, 06:22 AM #8 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 You're correct. The 234th is ABAJX. The 6112737th is PAPER.
 October 29th, 2018, 10:38 AM #9 Member   Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1 5 days of trying, but could not find the right way

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