The six sides of a convex hexagon A₁ A₂ A₃ A₄ A₅ A₆ are colored red. Each of the diagonals of the hexagon is colored either red or blue. If N is the number of colorings such that every triangle A_i A_j A_k , where 1 l < j < k 6 , has at least one red side, find the sum of the squares of the digits of N .

Step-by-step solution

Number of ways such that atleast one side of aligned A₂ A₄ A₆ is red & = ^3 C₁ 2^2- ^3 C₂ 2+ ^3 C₃ 2^0 & =7 aligned Number of ways such that atleast one side of A₁ A₃ A₅ is red =7 Number of ways to colour diagonals A₁ A₄, A₂ A₅ , aligned & A₃ A₆=2^3=8 & aligned Required number & =8 7 7 & =392= N Sum of square of digits & =3^2+9^2+2^2 & =94 aligned aligned