Take your constructed pyramid and hold it so that one stick only lies on
the table. Now, four sticks must branch off from it in different
directions--two at each end. Any one of the five sticks may be left out
of this connection; therefore the four may be selected in 5 different
ways. But these four matches may be placed in 24 different orders. And
as any match may be joined at either of its ends, they may further be
varied (after their situations are settled for any particular
arrangement) in 16 different ways. In every arrangement the sixth stick
may be added in 2 different ways. Now multiply these results together,
and we get 5 × 24 × 16 × 2 = 3,840 as the exact number of ways in which
the pyramid may be constructed. This method excludes all possibility of
error.
A common cause of error is this. If you calculate your combinations by
working upwards from a basic triangle lying on the table, you will get
half the correct number of ways, because you overlook the fact that an
equal number of pyramids may be built on that triangle downwards, so to
speak, through the table. They are, in fact, reflections of the others,
and examples from the two sets of pyramids cannot be set up to resemble
one another--except under fourth dimensional conditions!