Problem based on concepts of Triangles as well as probability

Hello all, Can someone help in solving this problem? Suppose M is a set of triangles formed by using sides of lengths 1,2,4 & 7 units: whatis the probability that a triangle at random from set will be equilateral? This involves the concepts of both Proability and triangles as well. Thanks in advance.

Equilateral triangles:

Number of possible equilateral triangles = 4
Letâ€™s find number of triangles with 2 equal sides Let the equal sides be 1 : other side cant be 4 or 7 (Sum of two sides should more than or equal to the third side) , number of possible triangles = 1 If the equal sides are 2 : 7 cant be the third side, So total number of triangles = 2 If the equal sides are 4 â€“ number of triangles = 3 If the equal sides are 7 â€“ number of triangles = 3 So total number of bilateral triangles are = 9 Finally we will try to find the number of triangles with all different sides 7 canâ€™t be the side of any of the triangle. (The other two large sides 2 + 4 < 7) And we cant have any triangle with 1, 2, 4 So the total number of triangles = 13. Thus the probability that one of the triangles is equilateral is 4/13

Re: Equilateral triangles

I think we had missed 2 triangles from the total list of triangles. Beacause the answer is given as 4/11. No.of equilateral triangles - 4C1 = 4 No.of Scalene triangles - 4C1 * 3C1 = 12 No.of normal triangles - 4C3 = 4 Among the total number of triangles 9 are removed (Example (2,2,7) (1,1,4) ....) (But i was unable to trace the entire 9.) So, the remaining triangles are = 20-9 = 11. Therefore Probability = 4/11. Thanks.