Challenge Problem: PICK UP STICKS

[akrish1982]

Source: Challenge Problem

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?

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.

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.

1. 1-2-3 is another option which would not have a possible triangle. this is not considered in the above solution.

I enlisted all possible outcomes (without order)

I get 20 possibilities (not 64 as I thought order did not matter). In 7 cases triangles could not be formed. so, my probability was 7/20. What is wrong in my approach?

A B C
1 1 1 triangle
1 1 2 no
1 1 3 no
1 1 4 no
1 2 2 triangle
1 2 3 no
1 2 4 no
2 2 2 triangle
1 3 3 triangle
1 3 4 no
2 2 3 triangle
1 4 4 triangle
2 2 4 no
2 3 3 triangle
2 3 4 triangle
3 3 3 triangle
2 4 4 triangle
3 3 4 triangle
3 4 4 triangle
4 4 4 triangle

[tim]

unless you are 100% sure you understand every possible trap and all the math that could possibly be used on a problem of this type, you are better to just work with all the possibilities rather than try to consolidate them. in other words, if you assume order doesn't matter without a profound understanding of the underlying mathematics, you are taking a big risk. you are taking no risk at all by working with all possibilities (in this case, by ordering them separately). sure it's more work, but it's better to do more work than to get the problem wrong. :) try it the other way and let us know if you still have questions..

[tim.badley]

I'm confused why 3 + 3 + 3 + 6 + 6 + 3 = 30 and not 24.

Cheers. Good problem

[jlucero]

I'm confused why 3 + 3 + 3 + 6 + 6 + 3 = 30 and not 24.

Cheers. Good problem

Good catch! The answer is still correct, but it's missing one possibility that won't work: 1-2-3. There are six different ways that this could occur, so the question should read:

3 + 3 + 3 + 6 + 6 + 6 + 3 = 30