If you have $5$ bannanas and $5$ apples and two carrots, and the two carrots are lined up with the $10$ fruits at random, what is the probability there are exactly two apples and any number of carrots between the two carrots. Assume that all items are distinct objects.
Attempt: So there are $10!$ ways to order the $5$ bannanas and $5$ apples. For each ordering, you can place the carrots in $\binom {11}{2}$ slots. So the denominator is $10!*\binom {11}{2}.$
Now we have to find all the points in the sample space where there exactly two apples, and any number of carrots between the two carrots. This is where I am stuck.