First off, the correct answer is $$584,640 = {10!\over 2!2!}- \left[{9! \over 2!}+{9! \over 2!}\right] + 8!$$ which can be found using the inclusion-exclusion principle.
My own approach is different from the above: In the word CALIFORNIA, we have 2 repeating A's and 2 I's, and 6 remaining unrepeated letters. We first place the 6 unrepeated letters, a total of 6! arrangements. Then, to avoid the A's and B's in consecutive positions, we place the 2 A's and 2 B's between the 6 letters, including the beginning and ending positions, which gives us 8 possible positions. The number of possible arrangements is the permutation of 7 out of 4, but we have to divide out the repeating A's and B's, which is $${7!\over 2!2!3!}$$ So in total, we have $${6!7!\over 2!2!3!} = 151,200$$ which is obviously different from the correct answer. Why is this wrong, and if possible, how I can fix this using the same approach?
Best Answer
We modify your approach by first placing the A's, then placing the I's.
We can arrange the six distinct letters C,L,F,O,R,N in $6!$ ways. This creates seven spaces, five between successive letters and two at the ends of the row.
Case 1: We choose two of these seven spaces in which to place the two A's, thereby separating them.
We now have eight letters. This creates nine spaces, seven between successive letters and two at the ends of the row. We choose two of these nine spaces for the I's. The number of such arrangements is $$6!\binom{7}{2}\binom{9}{2} = 544,320$$
Case 2: We place both A's in the same space.
We again have eight letters. This again creates nine spaces. The space between the two A's must be filled with an I. Therefore, there are eight ways to choose the position of the other I. The number of such arrangements is $$6!\binom{7}{1}\binom{8}{1} = 40,320$$
Total: These two cases are mutually exclusive. Hence, the total number of arrangements of the letters of the word CALIFORNIA in which no two consecutive letters are the same is $$6!\left[\binom{7}{2}\binom{9}{2} + \binom{7}{1}\binom{8}{1}\right] = 584,640$$ which agrees with the result obtained by using the Inclusion-Exclusion Principle.