In a competition, 5 teams' $2-2$ runners start. The team whose competitor finishes first gets 12 points. For the 2nd place, 7 points are awarded, for the 3rd place 4 points, for the 4th place 2 points, and for the 5th place 1 point. No points are awarded for further placements. Considering the points on the championship table, how many different outcomes are possible?
First, let's examine the cases where the club finishing in 1st place in the competition achieves the maximum number of points. This can be achieved if they secure both the 1st and 2nd places, in which case they have $12+7=19$ points. The club finishing in 2nd place in this scenario would have the maximum number of points if they secure the 3rd and 4th places, in which case their points are $4+2=6$, and the club finishing in 3rd place in the league is left with only 1 point. The second place could also have $4+1=5$ or just 4 points, but fewer is not possible, as even the points for the 4th and 5th places combined are less than those for the 3rd place. If the second place has 5 points, 2 points remain for the third place, and if the second place in the league only scores 4 points, the remaining two places are either secured by the third place or one of them (the one further back) is passed to the fourth place in the league.

In total, 4 league tables match the case where the leading club scores the maximum number of points. The following table lists the cases where the leading club secures the 1st place along with the 3rd, 4th, and 5th places, respectively. For better clarity, we also include the already examined case in the table.

| I. | $12+7$ |  |  |  | $12+4$ |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| II. | $4+2$ | $4+1$ | 4 | 4 | $7+2$ | $7+1$ | 7 | 7 |
| III. | 1 | 2 | $2+1$ | 2 | 1 | 2 | $2+1$ | 2 |
| IV. | - | - | - | 1 | - | - | - | 1 |

| I. | $12+2$ |  |  |  | $12+1$ |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| II. | $7+4$ | $7+1$ | 7 | 7 | $7+4$ | $7+2$ | 7 | 7 |
| III. | 1 | 4 | $4+1$ | 4 | 2 | 4 | $4+2$ | 4 |
| IV. | - | - | - | 1 | - | - | - | 2 |

The cases listed in the table are all different in the sense that for any two of them, there is a club whose points differ in these two cases.

Beyond the listed cases, additional cases are possible where the 1st club only secures the 1st place. In this case, it has 12 points, fewer is not possible, as the points for the 2nd and 3rd places combined are only 11. We count these cases using the following table. Another 10 cases are possible, so in total, there are 26 different outcomes.

| II. | $7+4$ |  | $7+2$ |  | $7+1$ |  | 7 |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| III. | $2+1$ | 2 | $4+1$ | 4 | $4+2$ | 4 | $4+2$ | $4+1$ |  |  |
| IV. | - | 1 | - | 1 | - | 2 | 1 | 2 | $2+1$ | 2 |
| V. | - | - | _ | - | - | - | - | - | - | 1 |