How Many Possible Fantasy Football Schedules Are There?

For a 12 team league that starts playoffs in week 15 there are around $1.72 * 10^{56}$ (172 Septendecillion) possible regular season schedules.

Assumptions

• There is no restriction on how many times two teams can play
  • For example, playing the same team every week can in a valid schedule
• You cannot play against your own team
• Leagues have a positive, even number of teams

Equation

• Number of possible schedules = $S_N$
• Number of teams in the league = $T_N$
• Number of weeks in regular season = $W_N$

$$S_N = (T_{N}-1)!!^{W_N}$$

info

The double explamation mark $!!$ is double factorial.

Calculating One Week

To show how this is calculated, let's start by looking at Week 1, $W_1$, in a 6 team league, $T_6$.

Let's say that Team 1, $T_1$, plays Team 2, $T_2$. Then there are 3 different ways for the rest of the matchups to go

Matchups
(1v2) (3v4) (5v6)
(1v2) (3v5) (4v6)
(1v2) (3v6) (4v5)

What if $T_1$ played against $T_3$? Then there are 3 more potential schedules for $W_1$

Matchups
(1v3) (2v4) (5v6)
(1v3) (2v5) (4v6)
(1v3) (2v6) (4v5)

Continuing with this process, we get a total of 15 possible schedules for $W_1$. This works regardless of which team is chosen to be $T_1$.

There were 5 groups of 3, or $5 * 3$. For an 8 team league, a similar pattern follows: $7 * 5 * 3 = 105$ possible schedules for a week. If we also multiply by $1$, for example $5 * 3 * 1$, this pattern is a double factorial, which is represented a double exclamation mark $!!$.

Therefore, the number of possible schedules in one week is $(T_{N}-1)!!$

$$(T_6 - 1)!! = 15 \qquad (T_8 - 1)!! = 105 \qquad (T_{10} - 1)!! = 945 \qquad (T_{12} - 1)!! = 10 395$$

info

For an 8 team league, $T_8 = 8$.

$(T_N - 1)!! = (T_8 - 1)!! = (8 - 1)!! = 7!! = 7 * 5 * 3 * 1 = 105$

10 team league: $9 * 7 * 5 * 3 * 1 = 945$

12 team league: $11 * 9 * 7 * 5 * 3 * 1 = 10 395$

Calculating Multiple Weeks

To get the number of schedules for a season, let's start by looking at the first two weeks of the season.

We already determined that the number of possible schedules in one week is $(T_{N}-1)!!$

There are 15 possible weekly schedules for a 6 team league. There are 15 different Week 1s and each of them could have 15 different Week 2s. So the total number of possible schedules for a 2 week regular season in a 6 team league is $15 * 15$, or $(T_{N}-1)!! * (T_{N}-1)!!$.

As we add more weeks to the schedule, we keep multiplying by 15, giving us 15 ^ Number of weeks in the regular season, or $15 ^ {W_N}$, or more generically $(T_{N}-1)!!^{W_N}$.