How many three-digit integers less than 601601 have exactly two different digits in their representation (for example, 232, or 466)?


Answer:

116

Step by Step Explanation:
  1. Let the two different digits be x and y.
    Therefore, the required integers are of the form xxy,xyx or yxx.
  2. 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,,6.
    Therefore, there are 6 integers with two zeros, i.e.100,200,,600.
  3. 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 9×3 =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 275=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+(5×22)=116.
  4. Hence, there are 116 three-digit integers less than 601 that have exactly two different digits in their representation.

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