A king has 1000 bottles of wine,A queen wants to kill the

Consider easier example: there are 7 bottles (1, 2, 3, ..., 7) out of which one is poisoned and there are only 3 prisoners (A, B, C) available. Is it possible to find out which bottle is poisoned?

Let:
A to taste bottle 1;
B to taste bottle 2;
C to taste bottle 3;
A and B to taste bottle 4;
A and C to taste bottle 5;
B and C to taste bottle 6;
A, B and C to taste bottle 7.

Now, if after 4 weeks only prisoner A will die we will know that bottle 1 is poisoned, if prisoners A and C will die we will know that bottle 5 is poisoned and so on.

If you look at the pattern above you'll see that it's basically listing all possible groups we can pick out of 3 prisoners: \(C^1_3=3\) - # of ways to choose 1 prisoner out of 3, \(C^2_3=3\) - # of ways to choose 2 prisoners out of 3, \(C^3_3=1\) - # of ways to choose 3 prisoners out of 3 --> \(C^1_3+C^1_3+C^1_3=7\).

There is a formula to find out how many different groups we can pick out of \(n\) distinct objects: \(C^0_n+C^1_n+C^2_n+...+C^n_n=2^n\), note that this number will include one empty group (\(C^0_n=1\)).

So if we have 3 prisoners then we can form \(2^3-1=7\) different groups with at least one prisoner in it (minus 1 as one group is empty group).

Back to the original question:

We have 1000 bottles. How many prisoners are needed to form enough different groups so that each group to taste different bottle? The death of some particular group will mean that the bottle this group is "assigned" to is poisoned.

\(2^n-1\geq{1000}\) --> \(2^n\geq{1001}\) --> \(n\geq{10}\). So minimum 10 prisoners are needed.

Answer: E.

P.S. If our aim is to minimize the number of deaths then 999 prisoners are needed and max number of deaths would be 1 (though it can be zero as well, if all these 999 prisoners will taste non-poisoned bottles).

By the way, i don't think it's GMAT question.

Hope it's clear.