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Q10 Placement Test II

A number is called normal if it has the property that its digits are not in strictly increasing or strictly decreasing order. How many 3-digit numbers are normal? (e.g. 366 is normal, but 367 is not).

Solution

Non-normal numbers are those that are either in increasing or decreasing order. Note that from 3 distinct digits, only one number can be formed in increasing order, and one number in decreasing order.

Now, for increasing order we can choose 3 digits in ⁹C₃ ways, and for decreasing order we can choose 3 digits in ¹⁰C₃ ways, since 0 is allowed as the unit digit in decreasing order.

From above, total 3-digit non-normal numbers = ⁹C₃ + ¹⁰C₃
= 84 + 120
= 204

Therefore, total normal 3-digit numbers = 900 − 204 = 696

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