# How Many Triangles Are There? Here’s How to Solve the Puzzle

Indeed, there are as some ways to create that triangle as there are methods to select strains 1, 2 and three. Jumble the digits each which manner: 123, 132, 213, 231, 312, 321. There are six potentialities; likewise, any triangle in the diagram might be created six potential methods.

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

Here’s the place math turns into highly effective. The similar process works for any variety of strains. How many 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 giant quantity: 2,025,189,100.

The similar calculation applies irrespective of what number of strains there are. Compare that method to brute-force counting, which isn’t solely laborious and error-prone however gives no manner 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 totally different colours right into a bag. Pull out three. How many various potential coloration mixtures are there? 20, after all.

That’s combinatorics, and it’s helpful for fixing issues of this sort. It comes with its personal notation, to simplify the strategy of calculating, and entails plenty of exclamation factors. The expression n! — “n factorial,” when stated aloud — describes the product of multiplying all 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 on.

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