Problem:
Three distinct segments are chosen at random among the segments whose endpoints are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
Answer Choices:
A. 715553
B. 572443
C. 143111
D. 10481
E. 286223 Solution:
Assume without loss of generality that the regular 12-gon is inscribed in a circle of radius 1 . Every segment with endpoints in the 12-gon subtends an angle of 12360k=30k degrees for some 1≤k≤6. Let dk be the length of those segments that subtend an angle of 30k degrees. There are 12 such segments of length dk for every 1≤k≤5 and 6 segments of length d6. Because dk=2sin(15k∘), it follows that d2=2sin(30∘)=1,d3=2sin(45∘)=2, d4=2sin(60∘)=3,d6=2sin(90∘)=2,
d1=2sin(15∘)=2sin(45∘−30∘)=2sin(45∘)cos(30∘)−2sin(30∘)cos(45∘)=26−2, and d5=2sin(75∘)=2sin(45∘+30∘)=2sin(45∘)cos(30∘)+2sin(30∘)cos(45∘)=26+2.
If a≤b≤c, then da≤db≤dc and the segments with lengths da,db, and dc do not form a triangle with positive area if and only if dc≥da+db. Because d2=1<6−2=2d1<2=d3, it follows that for (a,b,c)∈{(1,1,3),(1,1,4),(1,1,5),(1,1,6)}, the segments of lengths da,db,dc do not form a triangle with positive area. Similarly,
d3=2<26−2+1=d1+d2<3=d4d4<d5=26+2=26−2+2=d1+d3, and d5<d6=2=1+1=2d2
so for (a,b,c)∈{(1,2,4),(1,2,5),(1,2,6),(1,3,5),(1,3,6),(2,2,6)}, the segments of lengths da,db,dc do not form a triangle with positive area. Finally, if a≥2 and b≥3, then da+db≥d2+d3=1+2>2≥dc, and also if a≥3, then da+db≥2d3=22>2=dc. Therefore the complete list of forbidden triples (da,db,dc) is given by (a,b,c)∈{(1,1,3),(1,1,4), (1,1,5),(1,1,6),(1,2,4),(1,2,5),(1,2,6),(1,3,5),(1,3,6),(2,2,6)}.
For each (a,b,c)∈{(1,1,3),(1,1,4),(1,1,5)}, there are (122) pairs of segments of length da and 12 segments of length dc. For each (a,b,c)∈{(1,1,6),(2,2,6}}, there are (122) pairs of segments of length da and 6 segments of length dc. For each (a,b,c)∈{(1,2,4),(1,2,5),(1,3,5)}, there are 123 triples of segments with lengths da,db, and dc. Finally, for each (a,b,c)∈{(1,2,6),(1,3,6)}, there are 122 pairs of segments with lengths da and db, and 6 segments of length dc. Because the total number of triples of segments equals ⎝⎛(122)3⎠⎞=(663), the required probability equals