How many arrangements are there in the word PHILOSOPHY, if HISY letters are kept together?

Question

How many arrangements are there in the word PHILOSOPHY, if all the letters are used? Among those how many arrangements are there, if HISY letters are kept together?(note that HISY need not be in the same order)

My try

For first part I was able to find answer $=\frac{10!}{2!2!2!}=453600$

I'm having trouble solving second part.

What I already know is I can consider HISY as a single letter and calculate the arrangements i.e $\frac{7!}{2!2!}=1260$

But among this there are cases where two H letters are placed on either sides of ISY and so on. Please help me to continue this calculation. Thank you.

Answer 1

There are $4!=24$ ways to arrange the letters HISY within the block, so you need $\frac{7!}{2!2!}\times 4!=30240$. But as you noticed, this counts arrangements with "H[ISY]H" twice each. There are $\frac{6!}{2!2!}=180$ way to arrange this block with the other letters and $3!=6$ ways to arrange the letters ISY between the H's, so we subtract $180\times 6=1080$ from $30240$ for a final count of $29160$.

Answer 2

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I think that your try is actually right. Because you have locked in one H in the word HISY and thus there is only one H out there, so I think that your answer is actually right, and you don't got that problem that you were saying.