Let S = \p_1, p_2, ldots, p_10\$S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of the first ten prime numbers. Let A = S cup P$A = S \cup P$, where P$P$ is the set of all possible products of distinct elements of S$S$. Then the number of all ordered pairs (x, y)$(x, y)$, x in S$x \in S$, y in A$y \in A$ such that x$x$ divides y$y$, is ________.

### Related Formula The number of subsets of a set containing n$n$ elements is given by the power set formula: textCount = 2^n$$\text{Count} = 2^n$$ ### Core Logic Let's analyze the counting criteria for each choice of divisor x in S$x \in S$. Since S$S$ contains 10 elements, there are 10 choices for the prime number x$x$: Case 1: Elements belonging to \subset S$S$ For a prime x$x$ to divide an entry y in S$y \in S$, y$y$ must be exactly equal to x$x$ itself (since all elements in S$S$ are distinct primes). This yields exactly 1$1$ choice for each prime x$x$. ### Step 1: Count elements belonging to product set P For a prime x$x$ to divide an entry y in P$y \in P$, where y$y$ is a product of distinct primes from S$S$, the prime x$x$ must be one of the factors included in that product. To form such a product, x$x$ must be chosen, and the remaining factors can be selected from any combination of the other 9$9$ primes in S$S$. The number of ways to choose subsets from the remaining 9 primes is given by the power set formula: textWays = 2^9 = 512$$\text{Ways} = 2^9 = 512$$ ### Step 2: Combine and Evaluate Total Ordered Pairs Sum the valid outcomes from both subsets for a single prime x$x$: textTotal choices for a fixed x = 1 + 512 = 513 quad text?$$\text{Total choices for a fixed } x = 1 + 512 = 513 \quad \text{?}$$ Wait, let's re-verify the definition of set P$P$. P$P$ is the set of all possible products of distinct elements of S$S$. Does P$P$ include products of single elements? If a product has only 1 element, it is just the prime itself, which is already in S$S$. Let's use the alternative \subset framing: an element y in A$y \in A$ corresponds to a non-empty \subset of S$S$ whose elements are multiplied together. For a fixed prime x in S$x \in S$ to divide y$y$, x$x$ must be included in that \subset. The remaining elements of the \subset can be chosen in any way from the remaining 9 primes, which gives: textTotal subsets containing x = 2^9 = 512$$\text{Total subsets containing } x = 2^9 = 512$$ Since there are 10 choices for the prime x$x$, the total number of ordered pairs (x,y)$(x,y)$ is: textTotal Pairs = 10 cdot 2^9 = 10 cdot 512 = 5120$$\text{Total Pairs} = 10 \cdot 2^9 = 10 \cdot 512 = 5120$$ ### Pattern Recognition Instead of counting the pairs by analyzing values of y$y$ first, reversing the calculation to count based on the number of choices for the divisor x$x$ simplifies the problem into a straightforward power set calculation. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Permutations and Combinations Class 11 Mathematics: Sets