Combinatorics Circular Arrangement

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Homework Statement

A circular table is arranged so as to have 9 different robots occupy the table. If there are 5 different types of robots, what is the number of possible arrangements of these robots?

Homework Equations

The Attempt at a Solution

If it wasn't a circular table, the answer would be 5^9, I suppose. But since it is circular, there would be repetitions.

<1,2,3,4,5,6,7,8,9> is the same as <2,3,4,5,6,7,8,9,1> and so on.

So I think I need to find the number of repetitions, and subtract it at from 5^9.

There are 9 equivalent seating arrangements for each 'permutation'.
for example,

<1,2,3,4,5,6,7,8,9>
<2,3,4,5,6,7,8,9,1>
<3,4,5,6,7,8,9,1,2>
<4,5,6,7,8,9,1,2,3>
<5,6,7,8,9,1,2,3,4>
<6,7,8,9,1,2,3,4,5>
<7,8,9,1,2,3,4,5,6>
<8,9,1,2,3,4,5,6,7>
<9,1,2,3,4,5,6,7,8>

So, is the answer 5^9/ 9 ?

If yes, why isn't it a whole number?

 

[lanedance]

not quite, consider the case when all 9 places have the same robot type, linking the circle doe snot make this equivalent to any other arrangements and there will not by any repetitions, so you need to be a little more careful with counting repeated sequences