10 dots

Probability Level 3

Consider this regular triangular lattice:

If three dots are chosen at random, what is the probability that they will form the vertices of an equilateral triangle?

If the probability is given by a b \dfrac{a}{b} where a a and b b are coprime positive integers, express your answer as a + b a+b .

Image credit : http://wearekalia.blogspot.com


The answer is 9.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Jan 2, 2018

There are 15 ways to get an equilateral triangle:

  • 9 unit triangles
  • 3 triangles of side length 2
  • 1 triangle of side length 3
  • 2 triangles in the middle (like the one below)

And, there are ( 10 3 ) = 120 \binom{10}{3} = 120 ways to choose 3 dots.

Therefore, the probability is 15 120 = 1 8 \dfrac{15}{120} = \dfrac{1}{8}

1 + 8 = 9 1+8 = \boxed{9}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

can we generalize this at all for a larger triangle of dots?

j c - 3 years, 4 months ago

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Great question! Let's see, the first three are... 1, 5, 15... So let's see... How many for 15 dots?

Geoff Pilling - 3 years, 4 months ago

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