18. How many two-digit numbers can be formed from the five digits $1,2,3,4,5$ if no digit is repeated?
Solution. Since two-digit numbers differ from each other either by the digits themselves or by their order, the desired number is equal to the number of arrangements of two out of five elements: \(A_{5}^{2}=5 \cdot 4=20\). Therefore, 20 different two-digit numbers can be formed.

When finding the number of arrangements, we multiply \(n\) consecutive decreasing integers, i.e., we are short of ( \(m-n\) ) consecutive decreasing integer factors to form a complete factorial.

Therefore, the formula for the number of arrangements can be written as

\[
A_{m}^{n}=\frac{\overbrace{m(m-1)(m-2) \ldots(m-n)(m-n-1) \ldots 2 \cdot 1}^{(m \text { factors }}}{(m-n)(m-n-1) \ldots 2 \cdot 1}
\]

From this, considering that the numerator is equal to \(m\) !, and the denominator is equal to \((m-n)\) !, we can write this formula in factorial form:

\[
A_{m}^{n}=\frac{m!}{(m-n)!}
\]