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Seven birds live in a nest. They are very organized; each day three of the birds fly out in search of food. In n consecutive days, every pair of birds has been in exactly one of the n daily search parties. What is the value of n? To help you understand what this question is asking you, suppose the seven birds are A, B, C, D, E, F, and G. Birds A and B form one of the possible pairs. It is required that Birds A and B are two of the three birds flying out in search of food on one day, and one day only.

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2 Answers 2

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There are 7 birds so there are

${7\choose2}=21$ pairs of birds.

Since 3 birds fly every day,

$n = \frac{21}{3}=7$

One possible schedule for the birds (A,B,C,...,G) ensuring every pair of birds flies exactly once is:

Day 1: ABC
Day 2: ADE
Day 3: AFG
Day 4: BDF
Day 5: BEG
Day 6: CDG
Day 7: CEF

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There are 21 pairs of birds ($\frac{7\times6}{2\times1}$) and each day three pairs go out, so it must be seven days ($\frac{21}3$).

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