Permutations of the set $\{1, 2, 3, 4, 5\}$ fulfilling certain conditions - Combinations Theory

For #4, you can use Inclusion-Exclusion principle -

$|U| = 5!$ (The general case, "universe")

$|\bigcup \limits_{i=1}^n A_i|$ = $3\cdot4! - 3\cdot3! + 2!$ (we use it to avoid double counting)

By combining them we get,

$|U| - |\bigcup \limits_{i=1}^n A_i| = 5! - 3\cdot4! + 3\cdot3! - 2! = 64$