Question 106
Let $n \geqslant 2$ be a natural number and $M = \mathbb{N} \times \mathbb{N} \times \ldots \times \mathbb{N}$ ($n$ times $\mathbb{N}$), the set of $n$-tuples of natural numbers. For each $a = (a_{1}, a_{2}, \ldots, a_{n}) \in M$ denote by $d_{a}$ the number of pairs $(i, j)$, $i, j \in \{1, 2, \ldots, n\}$, for which $a_{i} - a_{j} = 1$.
Determine the maximum value of the number $d_{a}$ as $a$ ranges over the set $M$.
Determine the maximum value of the number $d_{a}$ as $a$ ranges over the set $M$.