280wk8_x4 - Balls and Urns Balls and urns problems are...

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Balls and Urns “Balls and urns” problems are paradigmatic. Many prob- lems can be recast as balls and urns problems, once we fgure out which are the balls and which are the urns. How many ways are there oF putting b balls into u urns? That depends whether the balls are distinguishable and whether the urns are distinguishable How many ways are there oF putting 5 balls into 2 urns? IF both balls and urns are distinguishable: 2 5 = 32 Choose the subset oF balls that goes into the frst urn Alternatively, For each ball, decide which urn it goes in This assumes that it’s OK to have 0 balls in an urn. IF urns are distinguishable but balls aren’t: 6 Decide how many balls go into the frst urn: 0, 1, . . . , 5 IF balls are distinguishable but urns aren’t: 2 5 / 2 = 16 IF balls and urns are indistinguishable: 6 / 2 = 3 1 What iF we had 6 balls and 2 urns? IF balls and urns are distinguishable: 2 6 IF urns are distinguishable and balls aren’t: 7 IF balls are distinguishable but urns aren’t: 2 6 / 2 = 2 5 IF balls and urns are indistinguishable: 4 It can’t be 7 / 2, since that’s not an integer The problem is that iF there are 3 balls in each urn, and you switch urns, then you get the same solution 2 Reducing Problems to Balls and Urns Q1: How many di±erent confgurations are there in Tow- ers oF Hanoi with n rings? A: The urns are the poles, the balls are the rings. Both are distinguishable. Q2: How many solutions are there to the equation x + y + z = 65, iF x, y, z are nonnegative integers? A: You have 65 indistinguishable balls, and want to put them into 3 distinguishable urns ( x , y , z ). Each way oF doing so corresponds to one solution. Q3: How many ways can 8 electrons be assigned to 4 energy states? A:
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280wk8_x4 - Balls and Urns Balls and urns problems are...

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