Permutations and Combinations( Miscellaneous)

Question

In how many distinguishable ways can the symbols a,a,a,b,c,d,e,f be permuted in groups of 4?

Answer 1

Setup:

Break into cases based on the number of $a$'s being used.

If exactly $n$ copies of $a$ are used in the string, pick which locations in the sequence they occupy in $\binom{4}{n}$ ways. Then, from left to right, pick which character occupies each remaining space.

For example, if exactly two $a$'s are used, pick where they go in $\binom{4}{2}$ ways. Then pick the left-most non-$a$ character in the string in $5$ ways (as there are five non-$a$ characters available to choose from) and then pick the remaining character in $4$ ways (as after picking the earlier character it is no longer available, leaving only $4$ left to choose from). This gives a total of $\binom{4}{2}\cdot 5\cdot 4$ different ways where exactly two $a$'s were used.

Sum over all possible values of $n$ to get the final count.