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

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- Jan 21st 2010, 07:29 PMjmedsycombinations
In how many ways can the letters in "mississippi" be arranged such that there are no consecutive s's?

- Jan 22nd 2010, 06:57 AMmathaddict
- Jan 22nd 2010, 07:09 AMGrandad
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.

So I reckon the answer is:

$\displaystyle \binom84\times\frac{7!}{4!2!}=7350$ ways.Grandad

- Jan 22nd 2010, 02:50 PMjmedsy