Question 15
Find the number of ordered $7$-tuples $(a_1, a_2, a_3, \ldots, a_7)$ having the following properties:
• $a_k \in \{1, 2, 3\}$ for all $k.$
• $a_1 + a_2 + a_3 + a_4$ $+ a_5 + a_6 + a_7$ is a multiple of $3.$
• $a_1a_2a_4$ $+ a_2a_3a_5$ $+ a_3a_4a_6$ $+ a_4a_5a_7$ $+ a_5a_6a_1$ $+ a_6a_7a_2$ $+ a_7a_1a_3$ is a multiple of $3.$
• $a_k \in \{1, 2, 3\}$ for all $k.$
• $a_1 + a_2 + a_3 + a_4$ $+ a_5 + a_6 + a_7$ is a multiple of $3.$
• $a_1a_2a_4$ $+ a_2a_3a_5$ $+ a_3a_4a_6$ $+ a_4a_5a_7$ $+ a_5a_6a_1$ $+ a_6a_7a_2$ $+ a_7a_1a_3$ is a multiple of $3.$