Combinatorial Geometry... Or is it Geometrical Combinatorics?

Probability Level 3

A 10-cm stick has a mark at each centimeter. By breaking the stick at two of these nine marks at random, the stick is split into three pieces, each of integer length. The probability that those three pieces could be the side lengths of a triangle is $\frac{a}{b}$ . Find $a+b$ .

The answer is 7.

2 solutions

Benjamin Wong
Feb 18, 2014

There are 36 ways to split the ruler, because you can do cuts at the 1st and 2nd notch, the 1st and 3rd notch, up to the 1st and 9th notch, then the 2nd and 3rd notch, up to the 2nd and 9th notch, the last one being the 8th and 9th notch, for a total of 8+7+...+1=36 ways.

There are only two combinations to make a triangle, 3-3-4 or 2-4-4, for a total of 6 permutations

6/36=1/6, so the required answer is seven

We can as well solve the denominator by selection:

$9C2 = 36$ ways to select two points

Now, the rest is same

Soumya Chakraborty - 7 years, 3 months ago

Why $9\choose2$ ? Where do $9$ and $2$ come from?

Ralph Anthony Espos - 7 years, 2 months ago

Well, there are 9 notches upon which a cut can be made, and 2 cuts are made. Without even needing to know n take k, you can think of the 36 as the result of 9 possibilities of the first cut, 8 for the second, then dividing the product of these by 2 because cutting one notch then a second will produce identical cuts as cutting the second then the first.

Arthur Conmy - 4 years ago

Perfect!

Finn Hulse - 7 years, 3 months ago

Finn Hulse
Feb 23, 2014

The hardest part of this is to find the denominator. We have to find numbers that add to 10. I will not go into it, I just listed them out. Then we look for ways that add up to 10 and work in the triangle inequality. We see that there are only 2 ways! The sides can be 2, 4, and 4 or they can be 3, 3, and 4. Because there are 3 ways to arrange each one, the total is $\frac{6}{36}$ . Simplifying, we find $\frac{1}{6}$ . Adding 1 to 6 gets 7 as the answer. If you put 19 as the answer, you forgot that you are allowed to rearrange the side lengths. ;)

Combinatorial Geometry... Or is it Geometrical Combinatorics?

The answer is 7.

2 solutions

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