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The 4 sticks in a complete bag of Pick-Up Sticks are all straight-line segments of negligible width, but each has a different length: 1 inch, 2 inches, 3 inches, and 4 inches, respectively. If Tommy picks a stick at random from each of 3 different complete bags of Pick-Up Sticks, what is the probability that Tommy CANNOT form a triangle from the 3 sticks?
First, make sure that you understand the problem. Essentially, Tommy picks three line segments at random. Each of the line segments could be 1, 2, 3, or 4 inches long. Then he is going to try to form a triangle. Some of the time, evidently, he will not be able to do so. The question is this: what is the probability that he cannot form a triangle from the three segments?
Recall, from your knowledge of geometry, the so-called “Triangle Inequality”: in any triangle, each side length must be less than the sum of the other two side lengths. This is simply another way of saying that the shortest path between X and Y is a straight line. If you have a triangle linking points X, Y, and Z, then the shortest way to get from X to Y is to go straight there, rather than take the detour through Z. You can also express the Triangle Inequality this way: each side length must be more than the absolute difference of the other two side lengths.
Since there aren’t tons of options for the side lengths, let’s go ahead and start constructing cases that would fail the test.
1-1-2: These three lengths would not form a triangle, because the third side (2) should be less than the sum of the other two sides (1 + 1). Now we can count the rearrangements: there are 3 ways to rearrange 1-1-2 (in other words, Tommy could pick the 2-side first, second, or third). You can do this count manually (1-1-2, 1-2-1, or 2-1-1), or you can divide 3! by 2! (the repeats) to get 3 options.
1-1-3: Another 3 options that fail the test.
1-1-4: Another 3 options.
1-2-4: Another 6 options, because you can rearrange 3 distinct sides in 6 (= 3!) different ways.
1-3-4: Another 6 options.
2-2-4: Another 3 options.
These are all the possibilities for triples that don’t form triangles (make sure you don’t double-count). Adding up all the options, you get 3 + 3 + 3 + 6 + 6 + 3 = 30.
Finally, you have to divide by all the possible outcomes. Tommy has 4 outcomes in each bag, and he picks from 3 different bags. So he has 4 × 4 × 4 = 64 possible outcomes.
30/64 = 15/32.
The correct answer is C.
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