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.
Best Answer
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 $29160$.