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A is a set containing n elements. A subset P_1 is chosen and A is reconstructed by replacing the elements of P_1. The same process is repeated for subsets P_1, P_2, \ldots, P_m \: with \: m>1. The number of ways of choosingP_1, P_2, \ldots, P_m,P_1 \cup P_2 \cup \ldots \cup P_m=A so that P_1 \cup P_2 \cup \ldots \cup P_m=A is

 

Option: 1

\left(2^m-1\right)^{m n}
 


Option: 2

\left(2^n-1\right)^m


Option: 3

{ }^{m+\pi} C_m

 


Option: 4

none of these


Answers (1)

best_answer

Let A=\left\{a_1, a_2, \ldots, a_n\right\} for\: each \: a_i(1 \leq i \leq n)\\Either \: a_i \in P_j \: or \: a_i\notin P_j(1 \leq j \leq m)
\therefore There \; are\; 2^m choices\; in \; which\; a_i \; m\; a_j belongs\; to \; P_j,\\ Also \; there\; is \; exactly\; one\; choice,\; i\; e, a_i \notin P_j

\therefore a_i \in P_1 \cup P_2 \cup \ldots \cup P_m \text { in }\left(2^m-1\right) \text { ways. }
Since, there are n elements in the set A, the number of ways of constructing subsets.
P_1, P_2, \ldots, P_m \text { is }\left(2^m-1\right)^n


 

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Rishi

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