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There are five buckets, each of which can be open or closed. An arrangement is a specification of which buckets are open and which bucket are closed. Every person likes some of the arrangements and dislikes the rest. You host a party, and amazingly, no two people on the guest list have the same likes and dislikes. What is the maximum number of guests possible?

1. $5^{2}$
2. $\binom{5}{2}$
3. $2^{5}$
4. $2^{2^{5}}$

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There are $5$ buckets and each of them can be opened and Closed i.e. $2$ options for each bucket.

so number of arrangements of buckets $= 2*2*2*2*2 = 2^5 = 32$

Now for of these $32$ arrangement a person can either select the arrangement or reject the arrangement i.e. $2$ options for each arrangement.

So the number of ways in which every person can select a set of arrangements such that set of his arrangements selected are different from other person's selection

$= 2*2*2*2....32 \ times$

$= 2^{32}$

$= 2 ^{2^5}$

Hence Option $D.$ is the correct answer.