Question

How many natural numbers are there from $1$ to $1000$ which have none of their digits repeated.

Answer 1

From $1$ to $1000$ we have

1 digit, 2 digit and 3 digit numbers

1 digit$\to$ $9$ ways

2 digit

$\overline{\downarrow 9}\overline{9}$

cant take 0

$9\times 9=81$ ways

2nd digit can't take 1st digit number

similarly

3 digit

$\overline{9}\overline{9}\overline{8}$

$=9\times 9\times 8=648$

$=648+81+9=738$ ways