Problem 26 · 2015 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
A two-digit number with the digits x, y, can be written in the form \(\overline{xy}\). Let a, b, c be different digits. In how many ways can the digits a, b, c be chosen, so that \(\overline{ab} < \overline{bc} < \overline{ca}\)?
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Answer: A — 84
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Hint 1 of 2
All three of ab, bc, ca are genuine two-digit numbers, so a, b, c are each from 1 to 9 and distinct.
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Hint 2 of 2
Count ordered distinct triples (a, b, c) from 1–9 satisfying ab < bc < ca.
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Approach: count valid distinct digit triples
- Since ab, bc, ca are two-digit numbers, a, b, c are in {1, ..., 9} and all different.
- Among the 9·8·7 = 504 ordered distinct triples, count those with 10a+b < 10b+c < 10c+a.
- Careful counting gives 84 ways.
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