# combinations

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• Jan 21st 2010, 07:29 PM
jmedsy
combinations
In how many ways can the letters in "mississippi" be arranged such that there are no consecutive s's?
• Jan 22nd 2010, 06:57 AM
mathaddict
Quote:

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

sorry , grandad corrected .
• Jan 22nd 2010, 07:09 AM
Grandad
Hello jmedsy
Quote:

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.
Grandad
• Jan 22nd 2010, 02:50 PM
jmedsy
Quote:

Originally Posted by Grandad
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

The book agrees; thanks guys