Why is $\mathbb N \times \mathbb N$ not subset of $\mathbb N \times \mathbb N \times \mathbb N \times \mathbb Q \times \mathbb Q$? Why is $\mathbb N \times \mathbb N$ not subset of  $\mathbb N \times \mathbb N \times \mathbb N \times \mathbb Q \times \mathbb Q$ ?

Every element of $\Bbb N\times\Bbb N\times\Bbb N\times\Bbb Q\times\Bbb Q$ is an ordered $5$-tuple of the form $\langle n_1,n_2,n_3,q_1,q_2\rangle$, where $n_1,n_2$ and $n_3$ are in $\Bbb N$, and $q_1$ and $q_2$ are in $\Bbb Q$. An element of $\Bbb N\times\Bbb N$ is an ordered pair $\langle n_1,n_2\rangle$, where $n_1$ and $n_2$ are in $\Bbb N$. This isn’t even the right kind of object to belong to $\Bbb N\times\Bbb N\times\Bbb N\times\Bbb Q\times\Bbb Q$: it’s not an ordered $5$-tuple.
What is true is that $\Bbb N\times\Bbb N\times\Bbb N\times\Bbb Q\times\Bbb Q$ has many subsets that have natural bijections with $\Bbb N\times\Bbb N$. Some of them are:
$$\begin{align*}
&\Bbb N\times\Bbb N\times\{0\}\times\{0\}\times\{0\}\\
&\Bbb N\times\{0\}\times\Bbb N\times\{0\}\times\{0\}\\
&\{0\}\times\Bbb N\times\Bbb N\times\{0\}\times\{0\}\\
&\{0\}\times\{0\}\times\{0\}\times\Bbb N\times\Bbb N
\end{align*}$$
The last one works because $\Bbb N\subseteq\Bbb Q$.
