# Permutations of the word SUPERSTITIOUS

1. Feb 18, 2010

### Lainababe

1. The problem statement, all variables and given/known data

How many ways are there to rearrange the letters of the word “SUPERSTITIOUS” so that...
(a) ...each T is immediately preceded by an S?
For example, “STUERSTIPIOUS” has this property, while “SUPERSTITIOUS” does not.
(b) ...the R appears between the two Ts?
For example, “SUTERSPITIOUS” has this property, while “SUPERSTITIOUS” does not.

2. Relevant equations
I am having trouble figuring out which permutations to exclude and include, and was wondering what the rules would be on using the quotient rule to divide out unnecessary permutations.

3. The attempt at a solution
For (a) I know that there are 13! ways total with each letter being distinct. I think that I need to divide 13! by 2!2! because there's two groups where the switching of the letters doesn't matter, but I am not entirely sure. I also think that I need to divide out other non-distinct permutations, such as the identical 'i's, So my conclusion was that there will be 11!/2!2!2! ways.

For (b) I'm stuck. I know that there are 13! total ways with each letter being distinct, I know that i need to divide by each letters' mulitplicities, etc. I am very unsure about the combinations of the placements of the T's and R's though. Any ideas would be great!

Note: I don't want an answer. I kinda just want pushed in the right direction. =)

2. Feb 18, 2010

### dacruick

The concept for having letters always following another letter is to say that T and S are one letter. So you would have ST, ST, S, I, I, O, U, U, R, E, P. You have to be careful though, I'm not sure if the fact that there is another S in the mix that you could switch in and our the different S's and make more permutations. double check that but I dont think you need to take that into account. so you would have 11!/(2!*2!) since only the I's and U's are recurring.

and for b) do you mean that the R can be anywhere between the two T's, or just sandwhiched right between like TRT.