First slide

Permutations

Question

Let $A$ be a set containing $n$ elements. The number of ways of choosing two subsets $P$ and $Q$ of $A$ such that
$P \cap Q = ϕ$ is

Moderate

A

$2^{n}$
B

$3^{n}$
C

$4^{n} - 2^{n}$
D

$4^{n} - 3^{n}$

Solution

For each $x \in A$ , we have four choices;
(i) $x \in P, x \in Q$
(ii) $x \in P, x \notin Q$
(iii) $x \notin P, x \in Q$
(iv) $x \notin P, x \notin Q$
Out of these four choices, last 3 choices imply
$x \notin P \cap Q$
Thus, $P$ and $Q$ can be chosen in $3^{n}$ ways so that
$P \cap Q = ϕ$

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