June 6th, 2012, 05:26 AM  #1 
Member Joined: May 2012 Posts: 86 Thanks: 0  Counting Problem
A number is said to be strictly ascendent when every one of its digits is greater than the one before it (For example: 2348, 123, 349, 258 159,etc.). How many strictly ascendent numbers are there between 110000? Help is appreciated. 
June 6th, 2012, 05:55 AM  #2 
Senior Member Joined: Apr 2010 Posts: 215 Thanks: 0  Re: Counting Problem
Write out all digits in increasing order: We need 4 digits, so scratch 5 of them ( choices!), for example: Now we have 2 choices per digit, whether to scratch it or not. But we can't scratch all of them so we have choices in total for each 4digit number. So, the final answer is Edit: I think I might have counted some twice. 
June 6th, 2012, 06:36 AM  #3 
Member Joined: May 2012 From: Chennai,India Posts: 67 Thanks: 0  Re: Counting Problem
this seems to be a combinatorics problem... Valid digits are 1 5+4 5 6 7 8 9 If single digit nos can be counted as valid, then we have 9 numbers. For 12 to 89: If 1st digit is chosen as n, we have 9n choices (the next high numbers) to fix the second digit. So , it is 8+7+6+5+4+3+2+1 = 8* 9/2 = 36 numbers For 123 to 789: If 1st and 2nd digit are fixed as 12, we have 7 choice to fill the 3rd digit. If 1st and 2nd digit are fixed as 13, we have 6 choice to fill the 3rd so, it is (7+6+5+4+3+2+1)+(6+5+4+3+2+1)+....(2+1)+1 = 84 numbers For 1234 to 6789: iF first 3 digits are fixed as 123, we have 6 choices to fill 4th digit If firlst 3 digits are fixed as 134, we have 5 choices So, it is (6+5+4+3+2+1)+(5+4+3+2+1)+(2+1)+1= 56 numbers.. So we have totally 9+36+84+56 = 185 numbers. Any way to do this the number theory way? 
June 6th, 2012, 08:17 AM  #4  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Counting Problem Hello, Jakarta! Quote:
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June 6th, 2012, 08:23 AM  #5  
Senior Member Joined: Apr 2010 Posts: 215 Thanks: 0  Re: Counting Problem Quote:
 
June 6th, 2012, 09:18 AM  #6 
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Counting Problem I didn't think onedigit numbers would be considered "strictly ascendent". Are the digits of "5" in increasing order? 
June 7th, 2012, 04:05 AM  #7  
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16  Re: Counting Problem Quote:
 

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