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Show that if five integers are selected from the first eight positive integers, there must be a pair of these integers with a sum equal to 9.

can i choose same interger five times? ex. if i choose integer 1 five times.

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No you can't take  same integer

Now let us take set under consideration {1,2,3,4,5,6,7,8}

Possible pairs {{1,8},{2,7}{3,6},{4,5}}  (since (1,2)=(2,1) we have not taken it)

Now let us to make an set of 5 integers that won't have any of above pair

so A={1,2,3,4,X} and B={8,7,6,5,Y}

now for X and Y you have to take an element from another set which will ultimately give sum 9

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