how many different words of length n over set A = {c, d,e, f,g, a,h} with exactly k occurrences of the character c?

Question

how many different words of length n over set A = {c, d,e, f,g, a,h} with exactly k occurrences of the character c ? and how can i be able to solve those kind of problems ?

Answer 1

Think of the potential word of length $n$, as $n$ empty slots. You want to fill $k$ of them with the letter $c$ and the rest $n-k$ slots with the other $6$ letters.

In how many ways can you choose which of the $k$ slots to fill with $c$? This is exactly ${n\choose k} = \frac{n!}{k!(n-k)!}$.

After you place all the $c$'s, you need to decide what to put in each of the other $n-k$ remaining slots, and you have $6$ choices for each of them, so $6^{n-k}$ choices in total.

Thus, the total number of words is ${n\choose k} 6^{n-k}$.