I got a question yesterday, to count the number of triangles in the above figure. I counted them, then thought about a formula to do the counting. I can't do that. Can someone show me how?
Counting triangles
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$\begingroup$
combinatorics
geometry
discrete-mathematics
puzzle
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0`I counted them` It helped if you posted how you did that, since by the time you got it right for $n=5$ the formula should become apparent. – 2017-01-31
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1maybe $$5\times(5+4+3+2+1)=75$$ – 2017-01-31
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0@dxiv I just counted the normal way, but got 69. Must be wrong, though. – 2017-01-31
1 Answers
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Its rather easy, once you think about it.
Let $h$ be the number of horizontal lines, and $v$ the number of non-horizontal lines. To form a triangle, you need to choose two non-horizontal lines, and one horizontal as base.
You can choose non-horizontal lines in $^v\text C_2={{v(v-1)}\over{2}}$ ways. Any of the horizontal lines may be selected in $^h\text C_1=h$ ways.
So, the total number of triangles is ${{^v\text C_2}\cdot{^h\text C_1}}$
Which is ${{{v(v-1)}\over{2}}\cdot{h}}$
And thus
$${{hv(v-1)}\over{2}}$$
P.S. In your figure, it is ${{5×6×5}\over2}=75$