Combinatorics of Non-Repeating Digits

We count natural numbers with distinct digits up to 1000:

1. 1-digit numbers (1 to 9): $9$ numbers.

2. 2-digit numbers (10 to 99):

   • Tens digit: $9$ choices ($1$-$9$)
   • Units digit: $9$ choices ($0$-$9$ except tens digit) Total $= 9 \times 9 = 81$.

3. 3-digit numbers (100 to 999):

   • Hundreds digit: $9$ choices ($1$-$9$)
   • Tens digit: $9$ choices ($0$-$9$ except hundreds digit)
   • Units digit: $8$ choices ($0$-$9$ except previous two) Total $= 9 \times 9 \times 8 = 648$.

4. 4-digit number (1000): $1000$ has repeating digits, so $0$ numbers.

Total count $= 9 + 81 + 648 = 738$.