9. How many different ways are there to rearrange the letters in the word 'BRILLIANT' so that no two adjacent letters are the same after the rearrangement?
(1 mark)
有多少種不同的方法可把「BRILLIANT 」一字中的字母重新排列, 使得排列後沒有兩個相鄰的字母相同？
9. 55440
9. Ignoring the rule that no two adjacent letters be the same, the answer would be $\frac{9!}{2!2!}$ as there are 9 letters including two I's and two L's. From this we must count the number of permutations with two adjacent I's or two adjacent L's.
If the two I's are adjacent, we can treat them as one single letter and hence the number of permutations would be $\frac{8!}{2!}$. The same is true for permutations with two adjacent L's. There are overlappings between these two types of permutations though, as there are 7! permutations in which the two I's and the two L's are both adjacent. Hence the answer is
$$
\frac{9!}{2!\times 2!}-\left(\frac{8}{2!}+\frac{8}{2!}-7!\right)=55440 \text {. }
$$