Question

Which collection of three side lengths will not form a triangle

Answer 1

To construct a triangle, the sum of any 2 sides of the triangle should be greater than the third side of the triangle.
lets calculate the sums of all the given choices to find out which three lengths given do not form a triangle.

A.
 6,10,18
6 + 10 = 16 
16 is smaller than the third side-18
Therefore these 3 lengths do not form a triangle

B.10,12,15
10 + 12 = 22                  15+12 = 27                         15 + 10 = 25
22>15 (3rd side)            27>10 (3rd side)                    25>12 (3rd side)
In this triangle, sum of all the possible pairs of sides are greater than the third side therefore these lengths can form a triangle 

C.10,15,6
10 + 15 = 25                      15 + 6 = 21                              10+6 = 16
25 >  6                                 21>10                                      16>15
in this too, sum of all the possible pairs of sides are greater than the third side. These lengths can be used to form a triangle

D.10,18 ,12
10 + 18 = 28                      18 + 12 = 30                             10 +12 = 22
28> 12                                    30>10                                      22> 18
Therefore these lengths can also be used to form a triangle

correct answer is 6,10 and 18 these cannot form a triangle