cormen 7th edition counting [closed]

Question

How many ways are there for a horse race with three horses to finish if ties are possible?(Note: Two or three horses may tie)

My answer is 9 but correct answer is 13.

In the solution they have even considered the case of all three horses tying at 1st,2nd or 3rd position? How is it possible that in 3 horses race all three tie at third position? How can we get third without getting first and second?

Comments

No Tie = $6$

Winning Tie(All win) = $1$

Winning Tie(2 horses) = $3$

Losing Tie (2 horse) = $3$

$13$