# combinations

• 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
Quote:

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

• Jan 22nd 2010, 07:09 AM
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 $4$ s's, there are $\frac{7!}{4!2!}$ arrangements of the remaining $7$ letters, which have $4$ repeated i's and $2$ 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 $\binom84$ ways of doing this.

So I reckon the answer is:
$\binom84\times\frac{7!}{4!2!}=7350$ ways.
• Jan 22nd 2010, 02:50 PM
jmedsy
Quote:

Hello jmedsyIf, to begin with, we ignore the $4$ s's, there are $\frac{7!}{4!2!}$ arrangements of the remaining $7$ letters, which have $4$ repeated i's and $2$ 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 $\binom84$ ways of doing this.
$\binom84\times\frac{7!}{4!2!}=7350$ ways.