De-arrangements + Probablity

[sriharsha]

Bob is about to hang his 8 shirts in the wardrobe. He has four different styles of shirt, two identical ones of each particular style. What is the probability that no two identical shirts are next to one another?

The total number of arrangements would be 8!/2!2!2!2! right? But what about the numerator?

 

[pka]

Bob is about to hang his 8 shirts in the wardrobe. He has four different styles of shirt, two identical ones of each particular style. What is the probability that no two identical shirts are next to one another?
The total number of arrangements would be 8!/2!2!2!2! right? But what about the numerator?

Your total number calculation is correct.
Using inclusion/exclusion we get \(\displaystyle \displaystyle\sum\limits_{k = 0}^4 {{{\left( { - 1} \right)}^k} {\dbinom{4}{k}} \left[ {\dfrac{{(8 - k)!}}{{{2^{4 - k}}}}} \right]} \) (no two of the same style together).

 

[sriharsha]

Your total number calculation is correct.
Using inclusion/exclusion we get \(\displaystyle \displaystyle\sum\limits_{k = 0}^4 {{{\left( { - 1} \right)}^k} {\dbinom{4}{k}} \left[ {\dfrac{{(8 - k)!}}{{{2^{4 - k}}}}} \right]} \) (no two of the same style together).

Thanks!
I know its not supposed to be in this forum but where can i ask such an arrangement question?

In how many ways can i arrange 5 white balls and 5 red balls in a circle?