# How do you find the number of distinct arrangements of the letters in STREETS?

Feb 13, 2016

First, we can ask ourselves "How many arrangements can we make with 7 things (letters in this case)"

The answer is 7! (We can choose 7 in the first slot, 6 in the next, etc)

However, we must also take into account the repetition of some letters in the word STREETS.

We can do this by dividing the repetition.

The repetition occurs because we counted S in place of another S as a possible combination. However, this is not distinct.

What is wrong: "$S T R {E}_{1} {E}_{2} T S$" is one combination and so is $S T R {E}_{2} {E}_{1} T S$

What is right: $S T R {E}_{1} {E}_{2} T S$" and $S T R {E}_{2} {E}_{1} T S$ are not distinct, so we only count them once.

We divide the possible arrangements for EACH repeating letter.

Since there are 3 repeating letters: S twice, T twice, E twice,
we have

(7!) / (2!2!2!)

Divide it all out and we have 630 distinct combinations

Note: for these types of problems, we can simply do
total letters! / number of repeats of letter1 ! number of repeats of letter2 ! ... and so on for all repeating letters