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How many relations are there on the set {a, b, c, d}
that contain the pair (a, a)?

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Since there are 4 elements in the set, so the total possible relations on set is $2^{2^{4}}=2^{16}$

There are total 16 possible ordered pairs on the set.

Now since it is given that element (a,a) should be present in all the relations. So of the remaining 15 ordered pairs any of them can be either part of the relation or not.

$\binom{15}{0} + \binom{15}{1} + ... + \binom{15}{15}= 2^{15}$ 

So ans is 215

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