The idea is to pick the smallest possible 1st digit that will permit the number
to have 4 different digits.  Then we pick the smallest possible 2nd digit that
will permit the number to have 4 different digits, and so on until we have
picked all 7 digits.  

The smallest first digit we can pick is 1.  We can't pick 0 since it wouldn't have
7 digits.

The smallest 2nd digit we can pick is 0.  We have now picked 2 different digits,
and have 5 more digits to pick, and we only need to pick 2 more different
digits.

The smallest 3rd digit we can pick is also 0.  We have picked 2 different
digits, and have 4 more digits to pick, and we only need to pick 2 more
different digits.

The smallest 4th digit we can pick is also 0.  We have picked 2 different
digits, and have 3 more digits to pick, and we only need to pick 2 more
different digits.

The smallest 5th digit we can pick is also 0.  We have picked 2 different
digits, and have 2 more digits to pick, and we only need to pick 2 more
different digits.

The smallest 6th digit we can pick is 2.  We have picked 3 different digits, and
have 1 more digit to pick.

The smallest 7th and last digit we can pick is 3.

Answer: 1000023 is the smallest possible 7 digit number with 4 different digits.

Edwin