Question

How many three-digit integers less than 1000 have exactly two different digits in their representation (for example, 232, or 466)?

Answer 1

Answer:

There are 9 integers with two zeros, i.e. 100, 200, · · · , 800, 900. When the repeated digit is

non-zero, the integers have the form ‘xxy’, ‘xyx’ or ‘yxx’. If x = 1, y can be 0, 2, 3, 4, 5, 6, 7, 8

or 9, although we must ignore ‘011’ as this is a two-digit integer. This gives 26 different integers.

Similarly, there will be an additional 26 integers for every non-zero x value. Therefore, the

total number of three-digit integers less than 1000 that have exactly two different digits in their

representation is 9 + 9 × 26 = 243.