Side by Side

We observe that for every box in the lowermost row, each of it can be paired with every other box in the second row. Hence, we get number of pairings for every box in lowest row = $6 \times 4 = 24$ . Similarly, for every box in the middle row, each of it can be paired with any box from the topmost row, and we get the number of pairings for every box in the middle row = $4 \times 2 = 8$ . As there is no row above topmost one, there are $0$ possible pairings. Hence, the total number of pairings = $24 + 8 = 32$ .

That's only $(6 \times 4) + (4 \times 2) = \boxed {32}$

Python: 2.7:

1
2
3
4
5
6

rows = 3
count = 0
for row in xrange(1,rows):
    for column in xrange(row*2):
        count += (row + 1) * 2
print "Answer:", count