Question

How many different arrangements of the letters in the word PARALLEL are there?

Answer 1

We have 8 total letters, but \(A\) and \(L\) are repeated two and three times (respectively). Normally, we would have \(8\times7\times\cdots2\times1=8!\) possible arrangements, but we don't want repeats (presumably). This means that \(8!\) counts \(PARAL\color{red}{L}E\color{blue}{L}\) as distinct from \(PARAL\color{blue}{L}E\color{red}{L}\), which I'm assuming are not considered distinct.

To eliminate the repeats, consider the fact that we have \(2!\) ways to arrange the two \(A\)s and \(3!\) ways to arrange the \(L\)s. Removing these from the total \(8!\) count is a matter of division: \(\dfrac{8!}{2!3!}\)