Question 1196433
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The numbers are either 2 or 3 digits.  Logical analysis and observation of patterns makes it relatively easy to find the answer.<br>
(1) 2-digit numbers<br>
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.<br>
ANSWER part 1: The number of 2-digit numbers with a digit sum of 8 is 8*1 = 8.<br>
(2) 3-digit numbers<br>
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.<br>
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.<br>
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.<br>
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.<br>
ANSWER part 2: The number of 3-digit numbers with a digit sum of 8 is 8+7+...+2+1 = 36.<br>
FINAL ANSWER: The number of numbers between 10 and 1000 with a digit sum of 8 is 8+36 = 44.<br>