The six sides of a convex hexagon A_1 A_2 A_3 A_4 A_5 A_6 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

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Quantrex Academy · Accepted Answer

Correct answer: 94

The six sides of a convex hexagon A_1 A_2 A_3 A_4 A_5 A_6 are colored red. Each of the diagonals of the hexagon is

Number of ways such that atleast one side of

\begin{aligned} \triangle A_2 A_4 A_6 \text { is red } & ={ }^3 C_1 \times 2^2-{ }^3 C_2 \times 2+{ }^3 C_3 2^0 \\ & =7 \end{aligned}

Number of ways such that atleast one side of

\triangle A_1 A_3 A_5

is red

=7

Number of ways to colour diagonals

A_1 A_4, A_2 A_5

,

\begin{aligned} & A_3 A_6=2^3=8 \\ & \begin{aligned} \therefore \text { Required number } & =8 \times 7 \times 7 \\ & =392=\mathrm{N} \\ \therefore \text { Sum of square of digits } & =3^2+9^2+2^2 \\ & =94 \end{aligned}

\end{aligned}$

The six sides of a convex hexagon A_1 A_2 A_3 A_4 A_5 A_6 are...

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