How many three-digit integers less than ^@ 601 ^@ have exactly two different digits in their representation (for example, ^@ 232, ^@ or ^@ 466)? ^@
Answer:
^@ 116 ^@
- Let the two different digits be ^@ x ^@ and ^@ y. ^@
Therefore, the required integers are of the form ^@ xxy, xyx ^@ or ^@ yxx. ^@ - If the repeated digits are zero, we must ignore the form ^@xxy, xyx^@ as they will give us one and two digit numbers. ^@ Eg. 001, 010,^@ etc.
So, if ^@ x = 0, ^@ the integers have the form ^@ yxx ^@ and ^@ y ^@ can be ^@ 1, 2, 3, \ldots , 6. ^@
Therefore, there are ^@ 6 ^@ integers with two zeros, ^@ i.e. 100, 200, \ldots, 6 00.^@ - When the repeated digit is non-zero, the integers are of the form ^@ xxy, xyx ^@ or ^@ yxx. ^@
If ^@ x = 1, y ^@ can be ^@ 0, 2, 3, 4, 5, 6, 7, 8 ^@ or ^@ 9, ^@ therefore there are ^@9 \times 3^@ ^@= 27^@ possible integers but we must ignore ^@ 011 ^@ as this is a two-digit integer.
Since your number is less than ^@ 601 ^@ so we must ignore ^@ 611, 711, 811, 911. ^@
This gives ^@ 27 - 5 = 22 ^@ different integers.
Similarly, there will be an additional ^@ 22 ^@ integers for every non-zero value of ^@ x ^@.
Therefore, the total number of three-digit integers less than ^@ 601 ^@ that have exactly two different digits in their representation ^@= 6 + (5 \times 22) = 116. ^@ - Hence, there are ^@ 116 ^@ three-digit integers less than ^@ 601 ^@ that have exactly two different digits in their representation.
