How many permutations of {1,2,...,n}. How many permutations of {1,2,...,n} where n >=5 are there such that none of {1,2,3} are adjacent to one another?

Example: the permutation (5,3,1,4,2) does not meet the condition described in the problem because 1 and 3 are in two adjacent places and the permutation (2,5,3,4,1) as well as (1,6,2,5,3,4) meets this condition.

Consider $A$ representing a permutation of the elements $1,2,3$. Since the remaining $n-3$ elements may be permuted in $(n-3)!$ ways, the required number is $$n!-\binom{3}{2}2!(n-1)!+\binom{3}{3}3!(n-2)!$$ because of the inclusion-exclusion principle.
