A university’s mathematics department has 10 professors and will offer 20 different courses next semester. Each professor will be assigned to teach exactly 2 of the courses, and each course will have exactly one professor assigned to teach it. If any professor can be assigned to teach any course, how many different complete assignments of the 10 professors to the 20 courses are possible?[ "20!/2^(10)", "10!/2^9", "10^(20) - 2^(10)", "10^(20) - 100" ]

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the number of different complete assignments of the 10 professors to the 20 courses, we use the formula for permutations of $ n $ items taken $ r $ at a time, which is $ nPr = \frac{n!}{(n-r)!} $. Since each professor is assigned to teach exactly 2 courses, we consider the permutation of 20 courses taken 2 at a time. ‖ ‖ step 2 ⋮ The permutation of the first course is $ 20P2 = \frac{20!}{(20-2)!} = \frac{20!}{18!} $. This gives us the number of ways to assign the first course to a professor. ‖ ‖ step 3 ⋮ After assigning the first course, we have 18 courses left. The permutation for the second course is $ 18P2 = \frac{18!}{(18-2)!} = \frac{18!}{16!} $. ‖ ‖ step 4 ⋮ We continue this process until all courses are assigned. The total number of ways to assign all courses is the product of all these permutations: $ 20P2 \times 18P2 \times \ldots \times 2P2 $. ‖ ‖ step 5 ⋮ This product simplifies to $ \frac{20!}{18!} \times \frac{18!}{16!} \times \ldots \times \frac{2!}{0!} = 20! $. ‖ ‖ step 6 ⋮ Since each professor is assigned exactly 2 courses, we must divide by $ 2^{10} $ to account for the fact that the order of the 2 courses assigned to each professor does not matter. ‖ ‖ step 7 ⋮ The final calculation is $ \frac{20!}{2^{10}} $. Using the asksia-ll calculation list, we find that this equals 654729075. ‖ ∻Answer∻ ⚹ 654729075 ⚹ ∻Key Concept∻ ⚹Permutations and Combinations⚹ ∻Explanation∻ ⚹The problem is a permutation problem where we assign 20 different courses to 10 professors with each professor teaching exactly 2 courses. The order of assignment matters, but within each pair of courses assigned to a professor, the order does not matter, hence the division by $ 2^{10} $.⚹◊What is the formula to calculate the total number of ways to assign 10 professors to 20 courses such that each professor teaches exactly 2 courses and each course has exactly one professor assigned to it?⍭ Generate me a similar question◊

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