Let $S=\left\{p_1, p_2 \ldots ., p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs ( $x, y$ ), $x \in S$, $y \in A$, such that $x$ divides $y$, is ______.

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Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Let } \frac{y}{x}=\lambda \ & y=\lambda x \ & =10 	imes\left({ }^9 C_0+{ }^9 C_1+{ }^9 C_2+{ }^9 C_3+\ldots+{ }^9 C_9ight) \ & =10 	imes 2 \ & =10 	imes 512=5120\end{aligned}$

Let $S=\left\{p_1, p_2 \ldots ., p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$,…

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