How do we get $r^n$ as the number of divisions possible. Please give a full description.
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See http://math.stackexchange.com/questions/47345/number-of-ways-of-distributing-n-identical-objects-among-r-groups?rq=1, 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
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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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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.
Dennis Gulko
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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 $a_i$.
Inceptio
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