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?

Yes, you are correct. Use constructive counting. Begin by selecting the first letter in the string, which you said could be either $A$ or $C.$ There are $2$ ways to do this.
Now our problem becomes: construct a string of length $9$ with at least one $B.$ We count this with complementary counting - how many strings can we make without a single $B?$ We have $4$ choices for each letter, and we must select $9$ letters. There are a total of $5^{9}$ strings (without any restrictions). Therefore, the number of valid strings is $5^{9} - 4^{9}.$
Our final answer is $\boxed{2 \cdot \left(5^{9} - 4^{9}\right)}.$
