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June 6th, 2012, 06:26 AM   #1
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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 1-10000?

Help is appreciated.
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June 6th, 2012, 06:55 AM   #2
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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 4-digit number.

So, the final answer is

Edit: I think I might have counted some twice.
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June 6th, 2012, 07:36 AM   #3
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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 9-n 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?
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June 6th, 2012, 09:17 AM   #4
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Re: Counting Problem

Hello, Jakarta!

Quote:
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 1-10000 ?







[color=beige]. . [/color]



soroban is offline  
June 6th, 2012, 09:23 AM   #5
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Re: Counting Problem

Quote:
Originally Posted by soroban
Hello, Jakarta!

Quote:
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 1-10000 ?







[color=beige]. . [/color]



That's nice but you forgot single digits.
brangelito is offline  
June 6th, 2012, 10:18 AM   #6
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Re: Counting Problem


I didn't think one-digit numbers would be considered "strictly ascendent".

Are the digits of "5" in increasing order?

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June 7th, 2012, 05:05 AM   #7
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Re: Counting Problem

Quote:
Originally Posted by soroban
Are the digits of "5" in increasing order?
I think they are. A number is not ascendent if there are two digits x1 x2 appearing in that order and with x2<=x1. You cannot find two such digits in 5.
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