Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to [2010]

Question

Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to [2010]

Smart Achievers · Accepted Answer

(4)

$S = {1, 2, 3, 4}$

$Let A and B be disjoint subsets of S$

$Now for any element a \in S, it has three possibilities: either it is in A or B or none$

$\Rightarrow For every element out of 4 elements there are three choices$

$∴$ $Total options = 3^{4} = 81$

$Here A \neq B except when A = B = ϕ$

$∴$ $81 - 1 = 80 ordered pairs (A, B) are there for which A \neq B$

$Hence total number of unordered pairs of disjoint subsets = \frac{80}{2} + 1 = 41$

Want Us to Email you About Special Offers & Updates?