a) How many strings of 8 distinct letters can be made from the letters {a,b,c,d,e,f,g,h}?
b) How many of the strings you found in (a) do not have any of the elements of {a,b,c} next to each other?
The first part (a) is easy, it's just 8!. But I have no idea how to solve the second part. I understand what the question is asking, it wants all the permutations of the letters from the set {a,b,c,d,e,f,g,h} where ab, bc, ac, ba, cb, ca, cannot be in the permutation, but I still don't know what to do.
The answer is 5!(6 x 5 x 4).
For b, how about first we create string with $5$ distinct letters which are not $a,b,c$. This is similar to part a, but now it is $5!$.
Then we insert $a,b,c$ between the 5 letters, at the beginning of the string, or at the end. $a$ has 6 options, then $b$ has 5 options left, and $c$ has 4 options left.
$5!\times 6\times 5\times 4$