(related to Problem: Building The Tetrahedron)
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. Anyone 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 $$5 \times 24 \times 16 \times 2 = 3,840$$ as the exact number of ways in which the pyramid may be constructed. This method excludes all possibility of error.
A common cause of an 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!
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