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Question Number 207387 by sniper237 last updated on 13/May/24

L e t c a r d E = n, a n d t h e s e t o f p a r t s

S = {(A, B) \in P (E) \times P (E) / A \cap B = \emptyset}

S h o w t h a t c a r d S = 3^{n}

Answered by Berbere last updated on 13/May/24

i f c a r d (A) = k; E = A \cup \bar{A} t h e n u m b e r o f s u b s e t o f c a r d = k

i n E i s (\begin{matrix} n \\ k \end{matrix})

w e h a v d t o s h o o s e B i n (\bar{A}); c a r d (\bar{A}) = n - k

B \in P (\bar{A}) c a r d (P (\bar{A})) = 2^{n - k}

(A, B) c a n b e e c h o s e d b y \underset{k = 0}{\sum^{n}} (\begin{matrix} n \\ k \end{matrix}) 2^{n - k} = \underset{k = 0}{\sum^{n}} (\begin{matrix} n \\ k \end{matrix}) 1^{k} . 2^{n - k} = {(1 + 2)}^{n} = 3^{n}

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