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There are total 7C2 = 21 pairs eligible for symmetric relations.

These pairs are of form {(a,b) (b,a)}. This pair consists of 2 ordered pair (a,b) and (b,a).

Also ordered pair of form (a,a) is also eligible for symmetric relations. There are total 7 such pairs.

Now,

Total number of symmetric relations which contains exactly 4 ordered pair =

7C4 +    //4 ordered pairs of form (a,a)

21C2 +  //2 pairs of form {(a,b) (b,a)} which will give 4 ordered pairs

21C1 * 7C2  //1 pair of form {(a,b) (b,a)} and 2 ordered pairs of form (a,a)

= 686.

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