How many three-digit positive integers are there such that the tens digit is smaller than the hundreds digit but greater than the units digit?

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Correct answer marked below

120

Correct

144

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Explanation

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Official Explanation

Official

We have to select three distinct digits so that the Hundreds digit > Tens digit > Units digit.

This can be solved by applying the combination method:

Case 1: We have to select any three distinct digits among these nine digits for hundreds digit except zero. (Digits available: 1,2,3,4,5,6,7,8,9).

\binom{9}{3} = \frac{9 \times 8 \times 7}{3!} = 84

ways.…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.