5 black, 7 red, 9 blue, and 6 white marbles. I'm having trouble finding how many ways there is to arrange 5 black, 7 red, 9 blue, and 6 white marbles to find the probability that every white marble is adjacent to at least one other white marble.
If you could help me out that would be great!

There are, in general, $n!$ ways to arrange $n$ objects. So you'd have $(5+7+9+6)!$ but then as all the black marbles are identical, their permutations shouldn't be counted. For every 'good' permutation, you also have another $5!7!9!6!$ that only differ from it by permutations of same colored marbles. Thus the result is $\frac{(5+7+9+6)!}{5!7!9!6!}$.
For the probability, consider that whites must be in pairs or triples. This is equivalent to solving the previous problem for $3$ and $2$ white marbles. Notice that the two cases overlap when all $6$ marbles are in one group.
