2005 PMWC Problems/Problem T1

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

2005 PMWC (Problems)
Preceded by
Problem I15
Followed by
Problem T2
I: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
T: 1 2 3 4 5 6 7 8 9 10
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