44,  found using this one-line perl script



perl -e '$c=0; for($i=10;$i<1001;$i++) { $s = "$i"; @c = split(//,"$s"); $sum=0; foreach( @c ) { $sum += $_;};  if ($sum == 8) { print "$i   $sum\n"; } else { print "$i\n"; }}' | grep '   8' | wc -l



      44







To see the raw output of just the numbers that have digits adding to 8:



perl -e '$c=0; for($i=10;$i<1001;$i++) { $s = "$i"; @c = split(//,"$s"); $sum=0; foreach( @c ) { $sum += $_;};  if ($sum == 8) { print "$i   $sum\n"; }}'





And to see ALL the numbers considered:



perl -e '$c=0; for($i=10;$i<1001;$i++) { $s = "$i"; @c = split(//,"$s"); $sum=0; foreach( @c ) { $sum += $_;};  if ($sum == 8) { print "$i   $sum\n"; } else { print "$i\n"; }}'



 

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!
 


The numbers are either 2 or 3 digits.  Logical analysis and observation of patterns makes it relatively easy to find the answer.


(1) 2-digit numbers


The first digit can be any digit from 1 to 8 (8 possibilities); for each first digit there is only one digit that makes a digit sum of 8.


ANSWER part 1: The number of 2-digit numbers with a digit sum of 8 is 8*1 = 8.


(2) 3-digit numbers


The first digit can be any digit from 1 to 8.  For each of those digits, perform the analysis similar to that used for the 2-digit numbers.


1st digit 1: The sum of the other 2 digits must be 7, so the second digit can be any digit from 0 to 7.  That's 8 possibilities.


1st digit 2: The sum of the other 2 digits must be 6, so the second digit can be any digit from 0 to 6.  That's 7 possibilities.


You can do the detailed analysis for the other first digits if you want.  But it should be clear that the numbers of 3-digit numbers with other first digits will decrease by 1 for each increase of 1 in the first digit.


ANSWER part 2: The number of 3-digit numbers with a digit sum of 8 is 8+7+...+2+1 = 36.


FINAL ANSWER: The number of numbers between 10 and 1000 with a digit sum of 8 is 8+36 = 44.


 

Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
 .





The meaning of the word  " between "  in  Math problems is not precisely defined.



Every time you should explain,  if endpoints are included or not.



So,  I will re-formulate the problem in this form





    +---------------------------------------------------------------------+
    |   How many numbers from 10 to 999 inclusive have a digit sum of 8?  |
    +---------------------------------------------------------------------+




                                Solution





For a minute, I will change my problem and ask

    +-----------------------------------------------------------------------+
    |   How many numbers from 0 to 999 inclusive have a digit sum of 8?    |
    +-----------------------------------------------------------------------+


Then it is the same as to ask

   
    How many integer solutions does this equation have 

         +  +  = 8

    in integer non-negative numbers   >= 0,   >= 0,   >= 0 ?


The answer is known from Combinatorics ( " Stars and Bars method " ),

( see this Wikipedia article  

https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)#:~:text=In%20the%20context%20of%20combinatorial,his%20classic%20book%20on%20probability. )


The number of solution in this case is   =  =  = 45.


From it, we should subtract 1, which represents the number "8",
(we included it, when extended the set from [10,999] to [0,999]).


After subtracting, we get the final 


ANSWER.  There are 45-1 = 44  integer numbers from 10 to 999 (inclusive), such that the sum of their digits is 8.




Solved.







 

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