Combinatorics: How Many Ways Are to Arrange the Letters in VISITING?

[Shoney45]

Homework Statement

How many ways are there to arrange the letters of the word VISITING with no pairof consecutive I's?

Homework Equations

C(n,k) P(n,k)

The Attempt at a Solution

I am calculating the entire number of arrangements possible at P(8,5). I then want to find out the entire number of cases where there are at least two I's consecutively placed, and subract that from P(8,5).

Since there are eight places to put letters, there are seven possible ways to places to I's consecutively. What I am unsure of is if those seven arrangements of letters include the six possible cases where all three I's are consecutively placed.

 

[lanedance]

Based on the problem wording I would defintely consider cases where all three I's are consecutive - in fact exclusing these woul make the problem more difficult

I thin you could probably do this with cases without much difficulty

consider
IIxxxxxx
now there is 6! ways to choose the remaining letters, it doesn't matter where you put the 3rd I as you already have two consecutive Is.

now consider
xIIxxxxx
the only tricky part will be when you choose IIIxxxxxx, which has already been counted, so there is only 5.5! other ways to choose the other 6 letters

hopefully you can continue with this, note that I've assumed the Is are indistinguishable
so VISITING counts as one way to arrange the letters, even though were the Is distinguishable (or say the letters were drawn at random) they could be arranged 3! different ways

 

[Robert1986]

Based on the problem wording I would defintely consider cases where all three I's are consecutive - in fact exclusing these woul make the problem more difficult

I thin you could probably do this with cases without much difficulty

consider
IIxxxxxx
now there is 6! ways to choose the remaining letters, it doesn't matter where you put the 3rd I as you already have two consecutive Is.

now consider
xIIxxxxx
the only tricky part will be when you choose IIIxxxxxx, which has already been counted, so there is only 5.5! other ways to choose the other 6 letters

hopefully you can continue with this, note that I've assumed the Is are indistinguishable
so VISITING counts as one way to arrange the letters, even though were the Is distinguishable (or say the letters were drawn at random) they could be arranged 3! different ways

I think that I would count it by first determining how many ways you can arrange the 5 non-I's. There are 5! ways to do this. Now, look at the "diagram" below. The $'s represent non-I's and the %'s represent spaces between them:

%$%$%$%$%$%

Each of the percent symbols are valid locations for ONE I. Thus, for each of the 5! permutations of the non-I's, we just need to determine how many ways there are to place I's so that there are no to back to back. This is exactly the same as pick three of the percent signs. Since there is C(6,3) ways to do that, there must be C(6,3) ways to place the I's. So, if I have thought this through correctly, there are 5!C(6,3) ways to arrange the letters VISITING subject to the given constraints.

 

[lanedance]

yep that works well

 

[Shoney45]

Thanks you guys. That was all very helpful.