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355f2001_counting

# 355f2001_counting - M TH 355 N OTES cells n dist objects m...

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M TH 355 N OTES : Oct 18, 2001 Some Notes on Counting cells objects specification number notes n , dist m , dist Place all objects in the cells. Some cells may be empty. n m For each object choose a cell in n ways to put it in. n , dist m types Place exactly one object in each cell. Assume we have an unlimited num- ber of each type. m n For each cell we have m choices of type. n , dist m , dist m k objects in k th cell, k = 1 · · · , n . Here m = m 1 + · · · + m n . m ! m 1 ! · · · m n ! Place the objects in order, then permute the contents of each cell. n , indist m , dist, m = m 1 + · · · + m n m k objects in k th cell. Let p i , i = 1 , . . . , h be the number of oc- curences of each distinct value of the m k . m ! m 1 ! · · · m n ! 1 p 1 ! · · · p h ! Permute the cells having the same cardinality since we can’t distinguish them. n , dist, n m m , m = m 1 + · · · + m h m k objects of type k . Place all objects, at most one per cell. m ! m 1 ! · · · m h ! n m Choose successively the cells to receive an object of type k , in- cluding n m objects of “vir- tual” type. n , dist n , dist Place one object in each cell. n ! Think of the cells as being in or- der. n , dist m , dist, m n Place at most one object in each cell. n ! ( n m )! Introduce n m virtual distin- guishable object in ( n m )!

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355f2001_counting - M TH 355 N OTES cells n dist objects m...

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