Name for this type of problem: "determine the number of ways to arrange the letters of the word". [closed]

What are these type of questions called? I know if you have a word like tartar it would be. I just don't know what its called

$$\frac{6!}{2!\cdot 2!\cdot 2!} = 90$$

• I do not understand your answer since tatar has five letters. This class of problems is called permutations of a multiset. Nov 30, 2017 at 12:10
• @N.F.Taussig sorry I spelled the word wrong. I updated it. Nov 30, 2017 at 12:34

It is called "permutation with repetition".

There are three main groups of questions like this:

1) permutations: you have $n$ things and you would like to put them into $n$ boxes (with or without repetition). For example the number of ways $5$ people can sit on a straight bench or like your original question.

2) combinations: you have more things than boxes so you first have to choose them that are going to be placed and you do not care about the order you put these things, like how many ways you can dress up if you have $3$ different pairs of socks, $4$ different shirts and $5$ different pants, as you see here the order doesn't matter, just the result

3) variations: same as combinations but now the order is important, for example the number of licence plates of the form $\cdots-\star\star\star$ where the $\cdot$s can be letters and $\star$ can be any digit.

I am unsure if they are called exactly what I called them (these names I remember from high school in Hungary so the names can differ).

Observe that how they build upon eachother:

with permutation only the order is important since we are going to pick as many elements as boxes are, with combinations we have less boxes so we have to choose some elements first which we are going to put into the boxes but the order is not important and with variations even the ordering is important.