Question

How many positive integers less than 1000 have distinct digits?
Is there a way to approach through combinatorics?

Answer 1

HI!!

Answer 2

i would say they all have distinct values !

Answer 3

Oh, I'm sorry. Have distinct digits!

Answer 4

10 choices for the first digit, then 9 for the second and finally 8 for the third
counting principle from there

Answer 5

damn this could be hard..

Answer 6

oh wait.

Answer 7

1-digit integers: 1, ..., 9 - there are 9 integers.
2-digit integers: 10, ..., 99 - there are 90 integers, but in 9 of them (11, ..., 99) the
two digits are the same. So there are 90 − 9 = 81 2-digit integers with distinct digits.
Another way to get this answer is to consider the number of possibilities for each digit:
the first digit can be any non-zero digit, so it has 9 choices. The second digit can be any
digit except equal to the first one, so it has 9 choices too. There are 9 · 9 = 81 choices
total.
3-digit integers: 100, ..., 999 - there are 900 integers total, but some of them have a
repeating digit... The number of 3-digit integers with distinct digits can be counted as
follows: the first digit can be any non-zero digit, so it has 9 choices. The second digit
can be any digit except equal to the first one, so it has 9 choices too. Finally, the third
digit can be any digit except equal to the first digit or the second digit, so it has 8
choices. There are 9 · 9 · 8 = 648 choices total.
So there are 9 + 81 + 648 = 738 positive integers less than 1000 wit distinct digits.

Answer 8

http://mail.csufresno.edu/~mnogin/math114fall04/41-18-sol.pdf

Answer 9

or you could just read that ^ :)

Answer 10

You can use some combinatorics, but really just a little multiplication.

1000 does not have unique digits. It's out

Let's look at the rest - 0 - 999

* 1-9 : 9 total, all distinct
* 10-99 : 9*9=81 distinct, because 11,22,33,44,55,66,77,88,99 are removed (that is 90-9=81 distinct numbers). Note, in the MSD, 1-9 are allowed, in the second digit, 0-9 are allow, but we need to take out the digit that was used in MSD, leaving 9 digits for the LSD
* 100-999: 9*9*8=648 distinct, for similar reasons

Total = 9+81+648 = 738 distinct numbers

Answer 11

Thanks a bunch you all!