Possible results that can be in a horse race There are three horses: Uri, Uli and Buki.
Results that can be possible in the race are 13.


*

*Uri first, Uli second, Buki third.

*Buki first, Uri second, Uli third.

*Buki first, Uri and Uli second together.

*Uri and Buki first, Uli second.

*Uri, Buki and Uli first all together.



There are eight more possible results but didn't write them all because I guess you understand what I mean.
Given that information, they ask to calculate how many different possible results can be with 5 horses.
I can calculate how many results can be if all the horses come at different times:
$120 = 5\cdot4\cdot3\cdot2\cdot1$
How do I do it with the other cases? Please don't use complicated math, famous formulas or theoremas, I'm supposed to calculate this just with logic.
 A: If we are to use little machinery, we can divide into cases, count the number of ways for each, and then add up.
Roughly, the cases can be described as follows: $5$; $4$-$1$; $3$-$2$; $3$-$1$-$1$; $2$-$2$-$1$; $2$-$1$-$1$-$1$; $1$-$1$-$1$-$1$-$1$.
We count the number of possibilities for each.
$5$: This the the five-way tie. There is $1$ way this can happen.
$4$-$1$: Either $4$ tied first, and a loser, or $1$ winner, and $4$ tied for last, or to put it more nicely, for second. The horse by itself can be chosen in $5$ ways. It can be tied for first or tied for last, for a total of $(2)(5)$ possibilities.
$3$-$2$: Again we have $2$ possibilities, a two-way tie for first or for last. The group of $2$ can be chosen in $\binom{5}{2}=10$ ways, giving a total of $(2)(10)$. 
$3$-$1$-$1$: The group of $3$ can be chosen in $\binom{5}{3}=10$ ways. It can be in any  of $3$ places. The leftmost empty place can then be filled in $2$ ways, for a total of $(2)(3)(10)$.
$2$-$2$-$1$: The lone horse can be chosen in $5$ ways, and can be in any of $3$ positions. We can fill the leftmost remaining position in $\binom{4}{2}=6$ ways, for a total of $(6)(3)(5)$.
Only two to go, one of which you have done!
Remark: For a lot more information, please see the Online Encyclopedia of Integer Sequences.
