Find the number of distinguishable permutations that can be formed from the letters of the word PHILIPPINES.

Let's work through a simpler version of this, and then you can apply the technique to this problem.

How many distinct words can be made up from the letter AAABB? The number of ways that five distinct letters can be arranged is 5! But since there are 3 A's that means that there are 3! Duplicates of each arrangement of the A's, so you divide 5! By 3! Similarly since there are 2 B's you must also divide by 2! To eliminate duplicates. So the final solution is 5!/(3!2!) = 10. Spercifically they are:

AAABB
AABAB
ABAAB
BAAAB
AABBA
ABABA
BAABA
ABBAA
BABAA
BBAAA

Now, can you apply this same reasoning to your problem?