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2005 PMWC Problems/Problem T1

Revision as of 08:05, 9 October 2007 by 1=2 (talk | contribs) (New page: ==Problem== Call an integer "happy", if the sum of its digits is <math>10</math>. How many "happy" integers are there between <math>100</math> and <math>1000</math>? ==Solution== We hav...)
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Problem

Call an integer "happy", if the sum of its digits is $10$. How many "happy" integers are there between $100$ and $1000$?

Solution

We have a three digit integer in the form ABC. The number of happy integers per hundred is the number of positive integer solutions to $B+C=10-A$

For the one hundreds:

B+C=9: 10 solutions

two hundreds:

B+C=8: 9 solutions

so on, until there are two solutions in 9 hundreds.

$10+9+8\cdots+2=54$


See also

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