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# 18_Counting - Indistinguishable objects into...

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Indistinguishable objects into distinguishable boxes (IODB) Similar to combinations with repetitions. For example, consider the previously solved science fair problem. Three schools -- A, B and C -- are competing for a grand prize in a science fair competition. There are two judges. Each judge, anonymously, recommends one of the two schools. Calculate the number of ways the judges recommend the schools. Here, we can view n -- schools as distinguishable boxes r -- anonymous recommendations as indistinguishable objects n boxes => n-1 bars separate them r objects => when placed you have some sequence of r objects and n-1 bars Therefore, the problem is similar to constructing (n-1+r)-bit strings with exactly r zeros (objects). Theorem 5. The number of ways to place r indistinguishable objects into n distinct boxes is 10/25/11 How many terms are there in the expansion of (x+y) n ? How many terms are there in the expansion of (x+y+z)

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