How many 4-digit numbers have no repeating digits?

How many 4-digit numbers have no repeating digits?

If no zeros

9C4*4!

nCr(9,4)*4! = 3024

If there is a zero (zero cannot be the first digit)

9C3*3*3!

nCr(9,3)*3*3! = 1512

3014+1512 = 4526

How many 4-digit numbers have no repeating digits?

1.  $\begin{array}{|lcll|} \hline \{0,1,2,3,4,5,6,7,8,9\} \\ \text{4-digit numbers}: \\ \binom{10}{4} * 4! = \frac{10!}{6!} = 7*8*9*10 = 5040 \\ \hline \end{array}$

2.  $\begin{array}{|lcll|} \hline \{1,2,3,4,5,6,7,8,9\} \\ \text{3-digit numbers}: \\ \binom{9}{3}* 3! = \frac{9!}{6!} = 7*8*9 = 504 \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline 5040 \qquad - \underbrace{504}_{\text{zero cannot be the first digit}} = 4536\\ \hline \end{array}$