1. probability

the registration numbers of vehicles in certain country use digits 1 to 999. if a visitor has just entered the country, wht is the probbability tht the first car he sees has at least 2 identical digits. answer provided is (29/111). show me the soloution please

2. Hello, qweiop90!

The registration numbers of vehicles in certain country use digits 1 to 999.
If a visitor has just entered the country, what is the probbability that
the first car he sees has at least 2 identical digits?
There are 999 possible numbers.

The opposite of "at least two identical digits" is "no matching digits."
. . How many numbers have no matching digits?

There are: . ${\color{blue}9}$ one-digit numbers.

For two-digit numbers:
The first digit is not zero . . . there are 9 choices.
The second digit can be any digit except the first digit: 9 choices.
. . There are: . $9\cdot9 = {\color{blue}81}$ two-digit numbers.

For three-digit numbers:
The first digit is not zero . . . there are 9 choices.
The second digit can be any digit except the first digit: 9 choices.
The second digit can be any of the remaining 8 digits: 8 choices.
. . There are: . $9\cdot9\cdot8 = {\color{blue}648}$ three-digit numgers.

So there are: . $9 + 81 + 648 \:=\:{\color{blue}738}$ numbers with no matching digits.

Hence, there are: . $999 - 738 \:=\:{\color{blue}261}$ numbers with some matching digits.

Therefore: . $P(\text{some matching digits}) \;=\;\frac{261}{999} \;=\;{\color{red}\frac{29}{111}}$