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 twoway ties each of which could be a tie for first or last. 3*2=6.
1 threeway 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 twoway 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 20061128 15:11:31 