1. ## combinations

In how many ways can the letters in "mississippi" be arranged such that there are no consecutive s's?

2. Originally Posted by jmedsy
In how many ways can the letters in "mississippi" be arranged such that there are no consecutive s's?

3. Hello jmedsy
Originally Posted by jmedsy
In how many ways can the letters in "mississippi" be arranged such that there are no consecutive s's?
If, to begin with, we ignore the $\displaystyle 4$ s's, there are $\displaystyle \frac{7!}{4!2!}$ arrangements of the remaining $\displaystyle 7$ letters, which have $\displaystyle 4$ repeated i's and $\displaystyle 2$ repeated p's.

Now imagine that these 7 letters are arranged with a space in between each, and an additional space at each end. That's 8 spaces in all. 4 of these 8 spaces must be chosen to contain an s; and there are $\displaystyle \binom84$ ways of doing this.

So I reckon the answer is:
$\displaystyle \binom84\times\frac{7!}{4!2!}=7350$ ways.
Hello jmedsyIf, to begin with, we ignore the $\displaystyle 4$ s's, there are $\displaystyle \frac{7!}{4!2!}$ arrangements of the remaining $\displaystyle 7$ letters, which have $\displaystyle 4$ repeated i's and $\displaystyle 2$ repeated p's.
Now imagine that these 7 letters are arranged with a space in between each, and an additional space at each end. That's 8 spaces in all. 4 of these 8 spaces must be chosen to contain an s; and there are $\displaystyle \binom84$ ways of doing this.
$\displaystyle \binom84\times\frac{7!}{4!2!}=7350$ ways.