Question

Find the number of permutations in the word circus

Answer 1

Do you mean to ask, "how many distinct permutations of strings can we make from the letters in 'circus'?"

Answer 2

I believe it's \(\dfrac{6!}{2!}\).

There are 6 letters in 'circus,' so we're concerned with all possible strings of 6 letters. If all the letters were distinct, then you'd have 6 possibilities for the first letter, then 5 for the second, then 4 and so on, which gives you \(6!\) total strings/permutations.

However, there are two of the same letter (c). The \(2!\) in the denominator removes all the strings that are the same except for the location of the c's; here's an example:
\[\large\color{red}c~i~r~\color{blue}c~u~s\\
\large\color{blue}c~i~r~\color{red}c~u~s\]
The takeaway here is that \(6!\) double counts these sorts of permutations, so dividing by \(2!\) ignores these repeats because there are \(2!\) ways to place 2 of the same letter in alternating positions.