# Arranging Letters

• March 22nd 2010, 11:13 AM
FatherMike
Arranging Letters
How many ways can the letters a,a,a,a,b,b,b,b,c,c,c (4 a's, 4 b's, 3 c's) be arranged such that 4 consecutive letters are not the same?

Thanks
• March 22nd 2010, 12:47 PM
Plato
Quote:

Originally Posted by FatherMike
How many ways can the letters a,a,a,a,b,b,b,b,c,c,c (4 a's, 4 b's, 3 c's) be arranged such that 4 consecutive letters are not the same?

There are a total of $\frac{11!}{(4!)^2(3!)}$ ways to arrange the string of letters
Now count number of ways that all a's or all b's are together.
Hint: $|A|+|B|-|A\cap B|$ then substract.