1) In how many ways can letters of the word $\text{MAMMAL}$ be arranged in a line? 1) In how many ways can letters of the word MAMMAL be arranged in a line?
For this question I put
$$\frac{6!}{3!2!} = 60$$ since there are $3$ 'M's and $2$ 'A's

2) The letters of the word PROBABILITY are arranged at random. Find the probability that the two 'I' are separated. 
Do I have to group the two I's to find the probability of them being separated ? 
Thank you.
 A: Your answer to the first question is correct.  
You can determine the number of distinguishable arrangements of the word PROBABILITY using similar reasoning.  

 $$\frac{11!}{2!2!}$$

From these arrangements, we must exclude those in which the two I's are consecutive.  To do this, treat the two I's as a single letter.  We then have ten objects to arrange, one of which is the double I.  

 $$\frac{10!}{2!}$$

To find the number of distinguishable arrangements of PROBABILITY in which the two I's are separated, subtract the number of arrangements in which they are together from the total number of arrangements.

 $$\frac{11!}{2!2!} - \frac{10!}{2!}$$

A: For 2), you can first count the number of ways to arrange the letters other than the I's, 
$\hspace{.4 in}$ which is given by $\displaystyle\frac{9!}{2!}$ since there are 9 letters and two B's.
These 9 letters create 10 gaps, and there are $\dbinom{10}{2}$ ways to choose the two gaps for the I's.
Since there are $\displaystyle\frac{11!}{2!2!}$ ways to arrange the letters of PROBABILITY,
this gives a probability of $\displaystyle\frac{\frac{9!}{2!}\binom{10}{2}}{\frac{11!}{2!2!}}=\frac{\frac{9\cdot10!}{2!2!}}{\frac{11!}{2!2!}}=\frac{9}{11}$.
