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,…,6.
Therefore, there are 6 integers with two zeros, i.e.100,200,…,600. - 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 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+(5×22)=116. - Hence, there are 116 three-digit integers less than 601 that have exactly two different digits in their representation.