Let $S = \{p_{1}, p_{2}, . . ., p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$ , where P is the set of all possible products of distinct element 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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answer is 5120.

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Detailed Solution

Let $\frac{y}{x} = λ$
$\begin{array}{l} y = λx \\ = 10 \times (^{9} C_{0} +^{9} C_{1} +^{9} C_{2} +^{9} C_{3} + \dots . +^{9} C_{9}) \\ = 10 \times (2^{9}) \\ 10 \times 512 \\ 5120 \end{array}$

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