part a) How many strings in the letters a, b, and c have length 10
and exactly four a's?
I did $\binom{10}{4} = 210$ different ways for the strings of length $10$ in part a to be arranged, but I'm confused on the rest.
FIGURED OUT PART A, not part B yet
part b) How many strings in the letters a, b, and c have three a's and
at most six letters?
For strings of length $3$ there is only $1$ combo aaa. Then for strings of length of $4$ there are aaa, baaa, caaa, abaa, acaa, aaba, aaca, aaab, aaac so that is $8$, but I'm not sure of the pattern for lengths $5$ and $6$.
So now I have $1$ for length $3$, $8$ for length $4$.
Still having problems with finding lengths $5$ and $6$.
Best Answer
Hint: for the first part, how many choices are there for each of the remaining six positions? for part b, you use the same logic as for part a. So for six letters, you choose the three places for a's (how many ways), then choose the other three letters (how many ways?)