Number of angarams of BORBOTTIO in which same letters are close together

Question

I had the following problem:

We have the word BORBOTTIO. Find:

1. All the anagrams
2. All the anagrams that starts with BB
3. All the anagrams where the same letters are close together.

The first two questions are very easy. What about the third? I think $5!$ but in the book, the solution is $36$.

Could someone help me?

Answer 1

The only way to get the result is that the groups of $B´s, T´s$ and $O´s$ are ordered consecutively. Thus we have the block $\textbf B_2\textbf T_2\textbf O_3$ and the letters $R$ and $I$, where $B_2=BB, T_2=TT, O_3=OOO$

We can arrange $\textbf B_2,\textbf T_2,\textbf O_3$ in $3!=6$ ways. And $\textbf B_2\textbf T_2\textbf O_3,R,I$ can be arranged in $3!=6$ ways as well. So the answer is $6\cdot 6=36$