Permutations Practice Test 2

Let A = {${\mathit{a}}_1,{\mathit{a}}_2,...,{\mathit{a}}_{\mathit{n}}$}
For ${\mathit{a}}_1$ ∈ A, we have the following choices:
(i) ${\mathit{a}}_{\mathit{i}}$ ∈ P; ${\mathit{a}}_{\mathit{i}}$ ∈ Q
(ii) ${\mathit{a}}_{\mathit{i}}$ ∈ P; ${\mathit{a}}_{\mathit{i}}$ ∈ Q
(iii) ${\mathit{a}}_{\mathit{i}}$ ∈ P; ${\mathit{a}}_{\mathit{i}}$ ∈ Q
(iv) ${\mathit{a}}_{\mathit{i}}$ ∈ P; ${\mathit{a}}_{\mathit{i}}$ ∈ Q
Out of these only (ii), (iii) and (iv) imply ${\mathit{a}}_{\mathit{i}}$ ∉ P ∩ Q. Hence, the number of ways in which none of ${\mathit{a}}_1,{\mathit{a}}_2,...,{\mathit{a}}_{\mathit{n}}$ belong to P ∩ Q is $3^{\mathit{n}}$.