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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 Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,413 Thanks: 1024  
October 19th, 2018, 06:35 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,628 Thanks: 2077 
There are 26×25 = 650 twoletter words, the 234th being JI. There are 26×25×25 = 16250 threeletter words. etc. 
October 20th, 2018, 01:16 AM  #5  
Member Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1  Quote:
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: 20,628 Thanks: 2077 
Consider how many words precede JA and how many precede PABAB.

October 20th, 2018, 09:39 PM  #7 
Member Joined: Sep 2017 From: Saudi Arabia Posts: 37 Thanks: 1  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: 20,628 Thanks: 2077 
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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