SOLUTION: Consider the distinct arrangements of the 12 letters in the word PALEOZOOLOGY How many distinct arrangements have none of the 5 letters G, L, L, P, Z adjacent? I have no idea

First we'll find all the distinguishable arrangements of AEOOOOY.
Then we'll insert the others G,L,L,P,Z among them so that none will be
adjacent.

There are 7!/4! = 210 distinguishable arrangements of AEOOOOY.

This could be a sample of those, putting blanks for possible places to
insert other letters

_A_E_O_Y_O_O_O_

We can pick any 2 of the 8 blanks to put the 2 L's in C(8,2)=28 ways.

Then we'd have something like this, where the x's are places where the G,P,Z
cannot go, next to either L:

_A_ExLxOxLxO_A_O_Y_

So there are P(6,3) = 120 choices of positions for G,P,Z.

Answer: 210*28*120 = 705,600

Edwin