Question

Find the number of $10$ digit numbers using the digits $0,1,2,......9$ without repetition.

Answer 1

As repetition is not allowed so for the unit digit we have $9$ choice of numbers excluding $0$ and for the tenth digit number we have $9$ choice numbers excluding the number used in unit place ( including $0$), for then hundredth place we have $8$ choice of numbers,.... and so on.

If digit is not repeating then total number

$\begin{array}{c}=9\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =9\times 9!=9\times 362880\\ =3265920\end{array}$.