Letters Permutation and eliminate some option

Question

There are 5 letters A,B,C,D and E. How many permutations are possible of these 5 letters if AB , BC , CD & DC are not allowed ? I am very thankful to you if you solve this problem for me and tell me a proper way to solve it.

Answer 1

Let $P$ denote the set of all permutations.

Let $P_1$ denote the set of all permutations that contain $AB$ as substring.

Let $P_2$ denote the set of all permutations that contain $BC$ as substring.

Let $P_3$ denote the set of all permutations that contain $CD$ as substring.

Let $P_4$ denote the set of all permutations that contain $DC$ as substring.

Then to be found is:$$|P|-|P_1\cup P_2\cup P_3\cup P_4|$$

Evidently $|P|=5!=120$ and $|P_1\cup P_2\cup P_3\cup P_4|$ can be found by means of the principle of inclusion/exclusion.

Give that a try.