How Many Triangles Are There? Right here’s The best way to Resolve the Puzzle

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So now, divide out the redundancy. The full variety of triangles created by these six strains is (6×5×four)/6, or 20. That’s the reply.

Right here’s the place math turns into highly effective. The identical process works for any variety of strains. What number of triangles are created by seven nonparallel strains? That’s (7×6×5)/6, or 35. What about 23 strains? (23×22×21)/6, or 1,771. How about 2,300 strains? That’s (2300×2299×2298)/6, which is a massive quantity: 2,025,189,100.

The identical calculation applies irrespective of what number of strains there are. Examine that method to brute-force counting, which isn’t solely laborious and error-prone however supplies no approach to test the reply. Math produces the answer and the rationale for it.

It additionally reveals that different issues are, at coronary heart, similar. Put balls of six completely different colours right into a bag. Pull out three. What number of completely different doable coloration combos are there? 20, in fact.

That’s combinatorics, and its helpful for fixing issues of this kind. It comes with its personal notation, to simplify the method of calculating, and entails numerous exclamation factors. The expression n! — “n factorial,” when mentioned aloud — describes the product of multiplying all of the integers from 1 to n. So 1! equals 1; 2! equals 2×1, or 2; three! equals three×2×1, or 6. And so forth.

In the issue by Dr. Loh, the calculation for the variety of triangles could be rewritten like this: 6!/(three!three!).

It may be written as C(6,three), which is learn as “6 select three.” Extra broadly, it’s mathspeak for the variety of methods to decide on three objects out of 6. It’s generalized into this way:

C(n, r) = n!/((nr)!r!)

That’s the equation that college students memorize, the helpful shortcut. Give it a better look. The primary half — n!/(nr)! — is what captures the 6×5×four within the triangle calculation. The r! is what eliminates the redundancies.

Kenneth Chang


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