Bunuel wrote:
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?
A. 11/32
B. 13/32
C. 15/32
D. 17/32
E. 19/32
Kudos for a correct solution.
Another method would be to find the probability of making triangles.
Total number of ways of picking 3 sticks = 4*4*4 = 64
To make a triangle, sum of length of two sticks should be less than the length of the third stick or length of any one stick should be greater than the difference of the lengths of other two.
Equilateral triangle: Pick the same stick from all 3 bags. Number of triangles = 4
(1, 1, 1), (2, 2, 2) etc
Isosceles triangle (Only 2 sides same):
The two same sides cannot be 1 since their sum will be 2 and the third side will be either 2 or more.
If the same sides are 2 in length, the third side can be between 0 and 4 (exclusive). Third side can be 1 or 3.
If the same sides are 3 in length, the third side can be between 0 and 6 (exclusive). Third side can be 1, 2 or 4.
If the same sides are 4 in length, the third side can be between 0 and 8 (exclusive). Third side can be 1, 2 or 3.
This gives 8 triangles. Each of the 8 triangles can be selected in 3 ways e.g. (2, 2, 1) or (2, 1, 2) or (1, 2, 2)
Total number of triangles = 3 * 8 = 24
Scalene triangle (All sides different):
Any side cannot be 1 because the difference between the other two sides will be at least 1.
So the triangle must be 2, 3, 4. This can be selected in 3! = 6 ways
You can make a triangle in 34 ways so you cannot make it in 30 ways.
Probability = 30/64 = 15/32