The registration numbers of vehicles in a certain country use digits 1-999. If a visitor has just entered the country. What is the probability that the first car he sees has at least 2 identical digits?

Ans: 29/111

Assuming leading zeroes count as digits, then probability of seeing 3 distinct digits

= 10*9*8/1000
= 720/1000
=18/25
Probability of at least 2 identical digits
= 280/1000
=7/25

If leading zeroes are not displayed, then
there are 9*9*8=648 plates with distinct digits between 100-999
and 90 between 1 and 99
for a total of 738 out of 999 plates.
So probability of plates with at least 2 identical digits is
(999-738)/999
=29/11

Last line should read

29/111

How do you get 738/999??

To find the probability that the first car the visitor sees has at least 2 identical digits, we need to determine the total number of possible registration numbers and the number of registration numbers that have at least 2 identical digits.

Total number of possible registration numbers:
The registration numbers can have digits from 1 to 999, which means there are 999 possible registration numbers.

Number of registration numbers with at least 2 identical digits:
To calculate this, we can count the number of registration numbers that do not have at least 2 identical digits and subtract it from the total number of possible registration numbers.

Registration numbers with no identical digits:
There are 9 choices for the first digit (1-9), 9 choices for the second digit (excluding the digit already chosen), and 8 choices for the third digit (excluding the two digits already chosen). This gives us a total of 9 * 9 * 8 = 648 registration numbers with no identical digits.

Registration numbers with at least 2 identical digits:
To find the number of registration numbers with at least 2 identical digits, we subtract the number of registration numbers with no identical digits from the total number of registration numbers.
Total number of registration numbers - Registration numbers with no identical digits:
999 - 648 = 351

Therefore, there are 351 registration numbers with at least 2 identical digits.

Probability:
The probability is given by the number of favorable outcomes (registration numbers with at least 2 identical digits) divided by the total number of possible outcomes (total number of registration numbers).

Probability = Number of registration numbers with at least 2 identical digits / Total number of possible registration numbers:
Probability = 351 / 999

Simplifying the fraction:
Probability = 117 / 333

Therefore, the probability that the first car the visitor sees has at least 2 identical digits is 117/333 or approximately 0.351 or 35.1%.