# Arranging letters of a word so that two Letters are together

In how many ways can the letters of the word SOCKS be arranged in a line so that the two S's are together? In how many arrangements can the letters in SLOOPS be arranged so that the two O's are together?

I would think the answer to the first one would be: Treat the two S's as one entity and permute the letters: 4! and divide by 2! to account for the identical element S.Apparently not. However for the second question, you are able to use this method?

• For the first there is no need to divide by $2!$ because the two S's are combined entity. In the second case you need to divide by $2$ because of the two S's. Think if two S's were different, then $S_1LOOPS_2$ and $S_2LOOPS_1$ would be considered different. But they are same so you need to compensate for that overcounting by dividing by $2$. – Anurag A Dec 21 '16 at 19:34

For the first question, you are arranging $\fbox{SS}\text{OCK}$, where $\fbox{SS}$ is a single entity, so there are $4$ different elements and the permutations work out to $4!$ without further adjustment.
For the second question, you are arranging $\fbox{OO}\text{SLPS}$, where again $\fbox{OO}$ is a single entity, so there are $5$ elements of $4$ different values, one of which is repeated, and the permutations work out to $5!/2! = 60$, the division by $2!$ adjusting for the repeated S.
• If the Os were separable you would have $6$ letters ($6!$ options) but with two sets of 2-fold repeats (divide by $2!$, twice) for $6!/(2!2!) = 720/4 = 180$ options (if you don't care whether the Os are together or apart). Which gives the same answer (via $180-60$) of $120$ for requiring the Os to be separate. – Joffan Dec 21 '16 at 22:17