Let $$A = \{2, 3, 6, 7\}$$ and $$B = \{4, 5, 6, 8\}$$. Let $$R$$ be a relation defined on $$A \times B$$ by $$(a_1, b_1) R (a_2, b_2)$$ if and only if $$a_1 + a_2 = b_1 + b_2$$. Then the number of elements in $$R$$ is _________

$$A = \{2,3,6,7\}$$, $$B = \{4,5,6,8\}$$. Relation on $$A \times B$$: $$(a_1,b_1)R(a_2,b_2)$$ iff $$a_1+a_2 = b_1+b_2$$.

We need to count pairs $$((a_1,b_1),(a_2,b_2))$$ where $$a_1+a_2 = b_1+b_2$$, i.e., $$a_1-b_1 = b_2-a_2 = -(a_2-b_2)$$.

So we need $$a_1 - b_1 = -(a_2 - b_2)$$.

Let $$d = a - b$$ for each pair $$(a,b)$$. Possible values of $$d$$:

For each $$(a,b) \in A \times B$$: $$d = a - b$$ ranges over $$a \in \{2,3,6,7\}$$, $$b \in \{4,5,6,8\}$$.

All possible $$d$$ values and their counts:

$$d = 2-4=-2$$: (2,4); $$d = 2-5=-3$$: (2,5); $$d = 2-6=-4$$: (2,6); $$d = 2-8=-6$$: (2,8)

$$d = 3-4=-1$$: (3,4); $$d = 3-5=-2$$: (3,5); $$d = 3-6=-3$$: (3,6); $$d = 3-8=-5$$: (3,8)

$$d = 6-4=2$$: (6,4); $$d = 6-5=1$$: (6,5); $$d = 6-6=0$$: (6,6); $$d = 6-8=-2$$: (6,8)

$$d = 7-4=3$$: (7,4); $$d = 7-5=2$$: (7,5); $$d = 7-6=1$$: (7,6); $$d = 7-8=-1$$: (7,8)

For $$(a_1,b_1)R(a_2,b_2)$$: need $$d_1 = -d_2$$, or $$d_1 + d_2 = 0$$.

Count pairs with $$d_1 + d_2 = 0$$:

$$d = -6$$: 1 pair. Need $$d = 6$$: 0 pairs. Contribution: 0.

$$d = -5$$: 1 pair. Need $$d = 5$$: 0. Contribution: 0.

$$d = -4$$: 1. Need $$d = 4$$: 0. Contribution: 0.

$$d = -3$$: 2 pairs {(2,5),(3,6)}. Need $$d = 3$$: 1 pair {(7,4)}. Contribution: $$2 \times 1 = 2$$.

$$d = -2$$: 3 pairs {(2,4),(3,5),(6,8)}. Need $$d = 2$$: 2 pairs {(6,4),(7,5)}. Contribution: $$3 \times 2 = 6$$.

$$d = -1$$: 2 pairs {(3,4),(7,8)}. Need $$d = 1$$: 2 pairs {(6,5),(7,6)}. Contribution: $$2 \times 2 = 4$$.

$$d = 0$$: 1 pair {(6,6)}. Need $$d = 0$$: 1 pair. Contribution: $$1 \times 1 = 1$$.

$$d = 1$$: 2 pairs. Need $$d = -1$$: 2 pairs. Contribution: $$2 \times 2 = 4$$.

$$d = 2$$: 2 pairs. Need $$d = -2$$: 3 pairs. Contribution: $$2 \times 3 = 6$$.

$$d = 3$$: 1 pair. Need $$d = -3$$: 2 pairs. Contribution: $$1 \times 2 = 2$$.

Total = $$0+0+0+2+6+4+1+4+6+2 = 25$$.

The answer is $$\boxed{25}$$.