Example 3.4.5. Ipad Security.
Revisiting your ipad’s security, what happens if the order in which the digits are entered does not matter? If so, then you would be picking a combination of 4 digits without replacement from a group of 10 digits. Namely,
\begin{align*}
\frac{10!}{4!6!} & = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4 \times 3 \times 2 \times 1 \times 6!}\\
& = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}\\
& = \frac{5040}{24}\\
& = 210.
\end{align*}
Notice that the total number of options is much smaller when order does not matter.
Note that if you were allowed to reuse the digits then the number of possible outcomes would be
\begin{align*}
\frac{13!}{4!9!} & = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} \\
& = 715
\end{align*}
which once again is more since numbers are allowed to repeat.