Number of strings over a set $A$ How can I calculate the number of strings of length $10$ over the set $A=\{a,b,c,d,e\}$ that begin with either $a$ or $c$ and have at least one $b$ ?
Is it accomplished through some sort of combinatorial logic coupled with discrete mathematics?

$$(\text{Number of strings of length }9\text{ with at least one }b)\\ = (\text{Number of strings of length } 9)-(\text{Number of strings of length }9 \text{ without }b) \\ = 5^9-4^9$$
So that 
$$\text{Number of strings of length }10\text{ that begin with either }a\text{ or }c \text{ and have at least one }b \\= 2(5^9-4^9)$$
