Problem: in a party every person shakes hands with every other person . if there are 105 hands shakes , find the number of person in the party .
Rationale: "let n be the number of persons in the party . number of hands shake = 105 ; total number of hands shake is given by nc 2 . now , according to the question , nc 2 = 105 ; or , n ! / [ 2 ! * ( n - 2 ) ! ] = 105 ; or , n * ( n - 1 ) / 2 = 105 ; or , n 2 - n = 210 ; or , n 2 - n - 210 = 0 ; or , n = 15 , - 14 ; but , we can not take negative value of n . so , n = 15 i . e . number of persons in the party = 15 . option d"
annotated_formula: divide(add(sqrt(add(multiply(multiply(105, const_2), const_4), const_1)), const_1), const_2)