New Zealand
How many three-digit integers less than 601601601 have exactly two different digits in their representation (for example, 232,232,232, or 466)?466)?466)?
Answer:
116116116
- Let the two different digits be xxx and y.y.y.
Therefore, the required integers are of the form xxy,xyxxxy,xyxxxy,xyx or yxx.yxx.yxx. - If the repeated digits are zero, we must ignore the form xxy,xyxxxy,xyxxxy,xyx as they will give us one and two digit numbers. Eg.001,010,Eg.001,010,Eg.001,010, etc.
So, if x=0,x=0,x=0, the integers have the form yxxyxxyxx and yyy can be 1,2,3,…,6.1,2,3,…,6.1,2,3,…,6.
Therefore, there are 666 integers with two zeros, i.e.100,200,…,600.i.e.100,200,…,600.i.e.100,200,…,600. - When the repeated digit is non-zero, the integers are of the form xxy,xyxxxy,xyxxxy,xyx or yxx.yxx.yxx.
If x=1,yx=1,yx=1,y can be 0,2,3,4,5,6,7,80,2,3,4,5,6,7,80,2,3,4,5,6,7,8 or 9,9,9, therefore there are 9×39×39×3 =27=27=27 possible integers but we must ignore 011011011 as this is a two-digit integer.
Since your number is less than 601601601 so we must ignore 611,711,811,911.611,711,811,911.611,711,811,911.
This gives 27−5=2227−5=2227−5=22 different integers.
Similarly, there will be an additional 222222 integers for every non-zero value of xxx.
Therefore, the total number of three-digit integers less than 601601601 that have exactly two different digits in their representation =6+(5×22)=116.=6+(5×22)=116.=6+(5×22)=116. - Hence, there are 116116116 three-digit integers less than 601601601 that have exactly two different digits in their representation.
