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How do we get $r^n$ as the number of divisions possible. Please give a full description.

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See math.stackexchange.com/questions/47345/…, somewhere in the answers there is a description for why the formula for non-identical groups is $r^n$ and why it's different for when the objects are identical –  muzzlator Apr 17 '13 at 15:45
    
If the groups are not labelled, the answer is not $r^n$. If they are labelled, the first object can be put into any one of the groups ($r$ choices), and for every such choice the second object can be put into any one of the groups, and so on. –  André Nicolas Apr 17 '13 at 15:49
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marked as duplicate by muzzlator, Ross Millikan, Thomas, vonbrand, Amzoti Apr 17 '13 at 16:25

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2 Answers

Hint:

$(a_1,a_2, \dots, a_n) \to (1,2, \dots r)$

Since, you can have any number of objects in one box.

$(a_1) \to (1,2, \dots r) =r$ ways

$(a_2) \to (1,2, \dots r)=r$ ways

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$(a_n) \to (1,2, \dots r)= r$ ways

$(a_i) \to (1,2, \dots r)$ represents the possible slots for $a_i$.

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Assuming the groups are labeled $1,...,r$, think of your objects as $n$ people standing in a line. Each division then is a word in the letters $1,...,r$ of length $n$: give each guy a sign with his group number. Hence there are $r^n$ such partitions.

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