The 4 sticks in a complete bag of Pick-Up Sticks are all straight-line

MANHATTAN GMAT OFFICIAL SOLUTION:

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.

in the official solution from manhattan, are we missing the impossible combination 1,2,3 ?

"in the official solution from manhattan, are we missing the impossible combination 1,2,3 ?"

Yeah, I was thinking the same...

Solution:

The total number of ways to pick one stick from 3 different bags of 4 sticks is 4 x 4 x 4 = 64.

The lengths of sticks picked from the bags that can’t form a triangle are those for which the sum of the shortest lengths is not greater than the longest length. Those lengths (when arranged from shortest to longest) are:

1, 1, 2

1, 1, 3

1, 1, 4

2, 2, 4

1, 2, 3

1, 2, 4

1, 3, 4

However, for each of the first 4 sets above, there are 3!/2! = 3 ways to arrange the 3 numbers and for each of the last 3 sets, there are 3! = 6 ways to arrange the 3 numbers. Therefore, there are a total number of 4 x 3 + 3 x 6 = 12 + 18 = 30 ways that the 3 lengths can’t form a triangle; thus, the probability in question is 30/64 = 15/32.

Answer: C

karishma Can you please correct me ,
Normally it is "each side length must be less than the sum of the other two side lengths." , In your reasoning you wrote that "To make a triangle, sum of length of two sticks should be less than the length of the third stick"

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