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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