Understand the sets

Let $S=\{p_1,p_2,\dots,p_{10}\}$ be the first ten prime numbers.

So $|S|=10$.

The set $P$ consists of all possible products of distinct elements of $S$.

That means every element of $P$ is obtained by choosing a non-empty subset of $S$ having at least 2 elements, and multiplying its elements.

Hence each element of $A=S\cup P$ corresponds exactly to a non-empty subset of $S$:

• subsets of size $1$ give the elements of $S$,
• subsets of size $\ge 2$ give the elements of $P$.

Therefore, every non-empty subset of $S$ gives one unique element of $A$.

So, $|A|=2^{10}-1=1023.$

Interpret the divisibility condition

We need the number of ordered pairs $(x,y)$ such that:

• $x\in S$,
• $y\in A$,
• $x$ divides $y$.

Since $x$ is a prime from $S$, and each $y\in A$ is a product of distinct primes from some non-empty subset of $S$, we have:

$x\mid y \iff \text{the prime } x \text{ appears in the subset corresponding to } y.$

Count for a fixed }x$

Fix some prime $x\in S$.

To form a $y\in A$ divisible by $x$, the subset corresponding to $y$ must contain $x$.

Now among the 10 primes, one prime is fixed as included ($x$), and each of the remaining 9 primes may be either included or not included.

Thus the number of subsets containing $x$ is $2^9=512.$

Each such subset gives exactly one element $y\in A$ divisible by $x$.

Hence, for each fixed $x$, the number of valid $y$ is $512.$

Count all ordered pairs

Since there are 10 choices for $x\in S$, $\text{Total number of ordered pairs} = 10\cdot 2^9 = 10\cdot 512 = 5120.$

Final answer

$\boxed{5120}$

Comparison with stored answer

Stored correct answer = $5120$.

Our derived answer matches the stored answer.