how many ways are there to arrange $8$ pennies and $5$ nickels in a How can you arrange a line of $8$ pennies and $5$ nickels, so that no $2$ nickels are next to each other (pennies are indistinguishable and nickels are too)?
Answer: I set nickels and pennies next to each other so I get ${9 \choose 5}*8!*5!=609638400$. Is this correct?

If you first lay out the $8$ pennies, there are $9$ spaces where the nickels can go ($7$ spaces between adjacent pennies and $2$ spaces at the two ends).  So the answer is simply $9\choose5$.  You would multiply this by $8!5!$ only if the pennies and nickels were all distinguishable.
