First slide

Permutations

Question

A is a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen. The set $A$ is reconstructed by replacing the elements of $P .$ $A$ subset $Q$ of $A$ is again chosen. The number of ways of choosing $P a n d Q$ so that $P \cap Q = ϕ$ is

Difficult

A

$2^{2 n} -^{2 n} C_{n}$
B

$2^{n}$
C

$2^{n} - 1$
D

$3^{n}$

Solution

Let $A = {a_{1}, a_{2}, ......., a_{n}} .$ For $a_{i} \in A$ , we have the following choices
$(i) a_{i} \in P a n d a_{i} \in Q$ $(i i) a_{i} \in P a n d a_{i} \notin Q$

$(i i i) a_{i} \notin P a n d a_{i} \in Q$ $(i v) a_{i} \notin P a n d a_{i} \notin Q$

Out of these only (ii),(iii) and (iv) imply $a_{i} \notin P \cap Q .$

Therefore, the number of required subsets is $3^{n} .$

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