22. We use the notation $\overline{a b}$ for the two-digit number with digits $a$ and $b$. Let $a, b, c$ be different digits. How many ways can you choose the digits $a, b, c$ such that $\overline{a b}<\overline{b c}<\overline{c a}$ ?
A 84
B 96
C 504
D 729
E 1000
22. A Since the three digits $a, b, c$ are different and $\overline{a b}<\overline{b c}<\overline{c a}$, it is necessary that $a<b<c$. But this condition is also sufficient. Also note that none of $a, b, c$ are zero because they each represent the tens digit of a two-digit number.
There are 9 ways to pick a non-zero digit, 8 ways to pick a second (different) digit, and 7 ways to pick a third digit. These digits can be arranged in $3 \times 2 \times 1=6$ ways, but only one of these will be in ascending order (we need $a<b<c$ ). Hence there are $(9 \times 8 \times 7) \div 6=84$ ways to choose the digits $a, b, c$.