The twelve signs of the horoscope (AQUARIUS, ARIES, and so on) run a race along the Zodiac. In how many different ways can the race end, if ties are possible?

I haven't found a natural extension to 12 competitors.  But I do have results for n={1,2,3,4,5}
For 1 team there is 1 result.
For 2 teams there are 3 results A then B, B then A, A and B tied.
For 3 teams there are 13 results:
3!=6 with no ties.
There are 3 possible two-way ties each of which could be a tie for first or last.  3*2=6.
1 three-way tie.
6+6+1=13

For 4 teams there are 75 results:
4!=24 with no ties
There are 4C2=6 possible results of a single two-way tie which leaves 3 places.  6*3!=36.
There are 4C3=4 possible 3=way ties which could be for first or last.  4*2=8.
There are 4C2*2C2=6 possible two way ties.
There is 1 possible four way tie.
24+36+8+6+1=75

For 5 teams I found 931 results but I'm not sure about it.

Posted by Jer on 2006-11-28 15:11:31