What is the number of triples ( a , b , c ) of positive integers such that (i) a < b < c < 10 and (ii) a , b , c , 10 form the sides of a quadrilateral?

Step-by-step solution

aligned & a+b+c>10 & (a, b, c) can be aligned a b c 1 2 8,9 1 3 7,8,9 1 4 6,7,8,9 1 5 6,7,8,9 1 6 7,8,9 1 7 8,9 1 8 9 2 3 6,7,8,9 2 4 5,6,7,8,9 2 5 6,7,8,9 2 6 7,8,9 2 7 8,9 2 8 9 3 4 5,6,7,8,9 3 5 6,7,8,9 3 6 7,8,9 3 7 8,9 3 8 9 4 5 6,7,8,9 4 6 7,8,9 4 7 8,9 4 8 9 5 6 7,8,9 5 7 8,9 5 8 9 6 7 8,9 6 8 9 7 8 9 Total 73 cases