This issue is in the section of my textbook that deals with the inclusion-exclusion principle. I don't see how to apply it here. Any tips?
Determine the number of simple permutations of the nine digits 1,2,. . . ,9 in which blocks 12, 34, and 567 do not appear.
There are $9!$ permutations in all. If $1,2$ are adjacent, consider them as a block; then there are $8!$ permutations of the resulting objects ($7$ remaining numbers, one block). Likewise, if $5,6,7$ are adjacent, there are $7!$ permutations of the resulting objects ($6$ remaining numbers, one block). To count the permutations in which several of the patterns appear, join each of them into a block. The result is
$$ 9!-8!-8!-7!+7!+6!+6!-5!=283560\;. $$