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PrintSouth African Mathematics Olympiad Second Round
South Africa counting and probability
Problem
For how many positive three-digit numbers is the hundreds digit smaller than the units digit?
Solution
360
Systematic counting soon reveals a pattern.
In the 900s there are clearly no cases.
In the 800s there are 10: 809, 819, 829, ..., 899.
In the 700s there are 20: 708, 718, 728, ..., 798 709, 719, 729, ..., 799.
In the 600s there are 30: 607, 617, 627, ..., 697 608, 618, 628, ..., 698 609, 619, 629, ..., 699.
This pattern continues until the 100s, where there are 80.
The desired total is thus .
Systematic counting soon reveals a pattern.
In the 900s there are clearly no cases.
In the 800s there are 10: 809, 819, 829, ..., 899.
In the 700s there are 20: 708, 718, 728, ..., 798 709, 719, 729, ..., 799.
In the 600s there are 30: 607, 617, 627, ..., 697 608, 618, 628, ..., 698 609, 619, 629, ..., 699.
This pattern continues until the 100s, where there are 80.
The desired total is thus .
Final answer
360
Techniques
Counting two waysSums and products