HomieDolphine
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the answer: The final answer is (2N)!, which is equal to 1*2*3*...*(2N-1)*2N. the explanation: Firstly decide which guests get the white hats. we have 2N guests and N white hats, so there are 2NcN ways to do this, which is ((2N)!/(N!)^2). Now in order to define the circles, we will define two functions which are onto and one to one between the group of people with white hat and the group of people with black hat (make sure you understand why defining those functions gives all the possible circles arrangements and every arrangement is given exactly one time). The first function will be for every person with white hat to decide excactly which person with black hat will stand to his right, there are N! ways to define this function. The second function will be for every person with black hat the decide exactly which person with white hat will stand to this right, there are N! ways to define this function. so we have ((2N)!/(N!)^2)*(N!)^2=(2N)!. Final answer: there are (2N)! ways.
the answer: The final answer is (2N)!, which is equal to 1*2*3*...*(2N-1)*2N. the explanation: Firstly decide which guests get the white hats. we have 2N guests and N white hats, so there are 2NcN ways to do this, which is ((2N)!/(N!)^2). Now in order to define the circles, we will define two functions which are onto and one to one between the group of people with white hat and the group of people with black hat (make sure you understand why defining those functions gives all the possible circles arrangements and every arrangement is given exactly one time). The first function will be for every person with white hat to decide excactly which person with black hat will stand to his right, there are N! ways to define this function. The second function will be for every person with black hat the decide exactly which person with white hat will stand to this right, there are N! ways to define this function. so we have ((2N)!/(N!)^2)*(N!)^2=(2N)!. Final answer: there are (2N)! ways.
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