Number of distinguishable arrangements of the word INDOOROOPILLY with three different conditions

Question

I have the following three questions on a past final exam, I wanted to ask if I have done everything correctly. Thank you!

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How many distinguishable arrangements are there for the letters of the word INDOOROOPILLY?

This is a derrangement question: We have $\frac{13!}{4!2!2!}$

What about if the letters PY(in that order) were an inseparable pair?

Then we could let PY=Z and check derrangements of the word INDOOROOILLZ, we have $\frac{12!}{4!2!2!}$

What about if it had to end with $O$ and start with $Y$?

Well take these two letters out of the word, since they are assumed, and then take the derrangement for the remaining characters, e.g. INDOROOPILL $\frac{11!}{2!3!2!}$

Answer 1

Your derangement answers are correct.