Math Is Fun Forum

The Configuration For Testing Location In The Spreadsheet:

Line #42
Configuration #38
Combo Group E04
Total Number of Combos - 56

I believe the 1001 combinations are all the possible combinations without any parameters.

We'll get there. 

--Mitch

NP.  Lets see if I can explain.

At its core fantasy football is a math-based game based on the real-life production of NFL players. Each week you fill out a lineup by "starting" players at the various positions from a Roster that is picked / drafted according to the league settings. The statistics your starting players accumulate on the field (yards, touchdowns, etc.) contribute to their point total for the week. The point totals of all of the players in your starting lineup are tallied into your weekly score, and if you have a higher total than your opponent (another member of your league) you win that week! Players who you do not start are considered on your "bench." They'll still score points like everyone else, but those points will not be counted toward your weekly total.

A Roster is made up of players made up of different positions - QuarterBacks (QB), RunningBacks (RB), WideReceivers (WR), TightEnds (TE), PlaceKickers (PK) and Defense/Special Teams (TM).

In the scenarios we have been working with the Roster is comprised of:

QBs - 1 to 2
RBs - 3 to 6
WRs - 3 to 6
TEs - 2 to 4
PKs - 1 to 2
TMs - 1 to 2

At no time can a Roster have less than the minimum or maximum at any one position, nor can the Roster have less than 10 players or more than 14 players.

You came up with 46 different Roster combinations using the above settings - Posts #3 & #11

From this Roster of 14 Players an 'Owner/Manager' will choose which 10 players to start each week from the Roster and in this particular league they consist of:

1 QB
2 RBs
2 WRs
1 TE
1 PK
1 TM
2 Flex Players (either RBs, WRs or a TE).

Those players that aren't started are called the bench players.

In my posts #13 & #14 I stated that I calculated the starting lineup combination/possibilities for several of the Roster configurations using MIF Calculator and the spreadsheet shows the final tally of the total number of starting lineup combinations for each of the 46 Roster Combos.

There's a lot of other rules/nuances when playing in a fantasy football league, but for the time-being our focus is on how many different combinations of "Starters/Lineups" for each Roster Combination there can be and what is the grand total of the number individual combinations? But if you want to learn more, go ahead and do a Google search for "A Beginner's Guide To Fantasy Football" and in my particular case, you can read the Rules For The Invitational

-- Mitch

Yo Bob,

Thanks for your continued interest in my problem.

Now pull up a chair, take a deep breath, light up a cigarette, because what follows is quite lengthy.

++++++++++

Here's what I've been up to:

I've been manipulating the MIF combinations calculator to see if I could come up with a total for all the player lineup possibilities for each of the roster configurations - now at 46 with your last update.

Example 1 - Using the 14-man/position Roster Configuration of: 2 QB, 5 RBs, 3 WRs, 2 TEs, 1 PK, 1 TM

First, I calculated the number of different non-repetitive combinations when using qb1,qb2,te1,te2 which is 4.  [qb1,te1; qb1,te2; qb2,te1; qb2, te2)

Second, I then set up the MIF combinations calculator with the following:

Combinations without repetition (n=12, r=10)
Using Items: qb1,rb1,rb2,rb3,rb4,rb5,wr1,wr2,wr3,te1,pk1,tm1
Using Rule: has 2 of: rb1,rb2,rb3,rb4,rb5
Using Rule: has 2 of: wr1,wr2,wr3
Using Rule: has 1 of: qb1
Using Rule: has 1 of: pk1
Using Rule: has 1 of: tm1

And came up with the below 33 combinations:

List has 33 entries.
qb1,rb1,rb2,rb3,rb4,rb5,wr1,wr2,pk1,tm1
qb1,rb1,rb2,rb3,rb4,rb5,wr1,wr3,pk1,tm1
qb1,rb1,rb2,rb3,rb4,rb5,wr2,wr3,pk1,tm1
qb1,rb1,rb2,rb3,rb4,wr1,wr2,wr3,pk1,tm1
qb1,rb1,rb2,rb3,rb4,wr1,wr2,te1,pk1,tm1
qb1,rb1,rb2,rb3,rb4,wr1,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb3,rb4,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb3,rb5,wr1,wr2,wr3,pk1,tm1
qb1,rb1,rb2,rb3,rb5,wr1,wr2,te1,pk1,tm1
qb1,rb1,rb2,rb3,rb5,wr1,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb3,rb5,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb3,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb4,rb5,wr1,wr2,wr3,pk1,tm1
qb1,rb1,rb2,rb4,rb5,wr1,wr2,te1,pk1,tm1
qb1,rb1,rb2,rb4,rb5,wr1,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb4,rb5,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb4,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb5,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb3,rb4,rb5,wr1,wr2,wr3,pk1,tm1
qb1,rb1,rb3,rb4,rb5,wr1,wr2,te1,pk1,tm1
qb1,rb1,rb3,rb4,rb5,wr1,wr3,te1,pk1,tm1
qb1,rb1,rb3,rb4,rb5,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb3,rb4,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb3,rb5,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb4,rb5,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb2,rb3,rb4,rb5,wr1,wr2,wr3,pk1,tm1
qb1,rb2,rb3,rb4,rb5,wr1,wr2,te1,pk1,tm1
qb1,rb2,rb3,rb4,rb5,wr1,wr3,te1,pk1,tm1
qb1,rb2,rb3,rb4,rb5,wr2,wr3,te1,pk1,tm1
qb1,rb2,rb3,rb4,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb2,rb3,rb5,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb2,rb4,rb5,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb3,rb4,rb5,wr1,wr2,wr3,te1,pk1,tm1

Third, I multiplied the number of different combinations by 4 to come up with a grand total of 132 combinations (33 x 4)

++++++++++

Here's another example:

Example 2 - using the 14 man/position Roster Configuration of 2 QBs, 4 RBs, 3 WRs, 2 TEs, 2 PKs, 1 TM.

First, I calculated the number of different non-repetitive combinations when using qb1,qb2,te1,te2,pk1,pk2 which is 8. [ qb1,te1,pk1; qb1,te1,pk2; qb1,te2,pk1; qb1,te2,pk2; qb2,te1,pk1; qb2,te1,pk2; qb2,te2,pk1; qb2,te2,pk2 ]

Second, I then set up the MIF combinations calculator with the following:

Combinations without repetition (n=11, r=10)
Using Items: qb1,rb1,rb2,rb3,rb4,wr1,wr2,wr3,te1,pk1,tm1
Using Rule: has 1 of: qb1
Using Rule: has 2 of: rb1,rb2,rb3,rb4
Using Rule: has 2 of: wr1,wr2,wr3
Using Rule: has 1 of: te1
Using Rule: has 1 of: pk1
Using Rule: has 1 of: tm1

And came up with the following 7 combinations:

List has 7 entries.
qb1,rb1,rb2,rb3,rb4,wr1,wr2,te1,pk1,tm1
qb1,rb1,rb2,rb3,rb4,wr1,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb3,rb4,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb3,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb2,rb4,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb1,rb3,rb4,wr1,wr2,wr3,te1,pk1,tm1
qb1,rb2,rb3,rb4,wr1,wr2,wr3,te1,pk1,tm1

Third, I multiplied the number of the above different combinations to come up with a grand total of 56 combinations (7 x 8).

++++++++++

Notes:  The MIF calculator logic in its formula limits the number of 'has" and "no" rules to two or more.  It would have been a lot easier if it has a rule that required at 1 of 2 or more items.  i.e, 'must have, 1,qb1,qb2' Make sense?

I'm not sure your program is capable of the calculating above combinations without the additional steps that are involved with MIF's, but if it is, maybe, if you have the time and inclination, could you check my calculations to see if I'm on the right track / did it correctly?

Again thanks for all your input.

--Mitch

Sorry `about the confusion.

Roster is made up of the following (38 Combinations - Post #3):

QB:  Min = 1; Max = 2
RB:  Min = 3; Max = 6
WR: Min = 3; Max = 6
TE:  Min = 1; Max = 2
PK:  Min = 1; Max = 2
TM:  Min = 1; Max = 2

From the one of the configurations an owner would start a combination of 10 player consisting of:

QB - 1
RB - 2 (Max of 4)
WR - 2 (Max of 4)
TE - 1 (Max of 2)
PK -1
TM - 1
Flex Players - 2 (Either RBs, WRs, or TEs); there are six five combinations:  [ Because of the lineup limitations only 2 TEs can be started ]

1 QB, 2 RBs, 2 WRs, 3 TEs, 1 PK, 1 TM
1 QB, 2 RBs, 3 WRs, 2 TEs, 1 PK, 1 TM
1 QB, 2 RBs, 4 WRs, 1 TE, 1 PK, 1 TM
1 QB, 3 RBs, 2 WRs, 2 TEs, 1 PK, 1 TM
1 QB, 3 RBs, 3 WRs, 1, TE, 1 PK, 1 TM
1 QB, 4 RBs, 2 WRs, 1 TE, 1 PK, 1 TM

Which brings me back to the question "How many total different individual player combinations will there be for all the roster configurations with the starter limitations in place?  I'm thinking hundreds at least, possibly in the thousands?

For instance - some of the combinations for a different individual player starter configuration based on the roster makeup of: (Post #6)

QB = 2, RB = 6, WR = 3, TE = 1, PK = 1, TM = 1

For instance - some of the combinations for a different individual player starter configuration based on the roster makeup of:

QB = 2, RB = 6, WR = 3, TE = 1, PK = 1, TM = 1

Would be (Exchanged Player Red for Player Blue):

QB1
RB1, RB2, RB3, RB4
WR1, WR2
TE1
PK1
TM1

or

QB2
RB1, RB2, RB3, RB4
WR1, WR2
TE1
PK1
TM1

or

QB1
RB1, RB2, RB3, RB4
WR1, WR2
TE1
PK1
TM1

or

QB2
RB1, RB2, RB3, RB5
WR1, WR2
TE1
PK1
TM1

and more . . .

Make more sense Bob?

Thanks for your interest - Mitch

Follow-Up:  I'm thinking that to calculate the number of total combinations of Rosters vs. Lineups one would do what you did to calculate the number of Roster combinations, but do it for each of the different Roster configurations. But, what do I know? 

Alrighty then, can we take this to the next level?

We now know that there are 38 combinations based on the roster limitations of:

QB:  Min = 1; Max = 2
RB:  Min = 3; Max = 6
WR: Min = 3; Max = 6
TE:  Min = 1; Max = 2
PK:  Min = 1; Max = 2
TM:  Min = 1; Max = 2

The starters' configurations for a fantasy lineup is:

QB:  Min = 1; Max 1
RB:  Min =2; Max = 4
WR:  Min =2; Max = 4
TE:  Min =1; Max = 2
PK:  Min =1; Max = 1
TM:  Min =1; Max = 1

Can't start 0 at any position.

What would be the number of total individual starter player combinations for each roster combo?  I know that the there are only 6 5 starter configurations possible as listed below:

1 QB, 2 RBs, 2 WRs, 3 TEs, 1 PK, 1 TM
1 QB, 2 RBs, 3 WRs, 2 TEs, 1 PK, 1 TM
1 QB, 2 RBs, 4 WRs, 1 TE, 1 PK, 1 TM
1 QB, 3 RBs, 3 WRs, 1 TE, 1 PK, 1 TM
1 QB, 3 RBs, 2 WRs, 2 TEs, 1 PK, 1 TM
1 QB, 4 RBs, 2 WRs, 1 TE, 1 PK, 1 TM

For instance - some of the combinations for a different individual player starter configuration based on the roster makeup of:

QB = 2, RB = 6, WR = 3, TE = 1, PK = 1, TM = 1

Would be (Exchanged Player Red for Player Blue):

QB1
RB1, RB2, RB3, RB4
WR1, WR2
TE1
PK1
TM1

or

QB2
RB1, RB2, RB3, RB4
WR1, WR2
TE1
PK1
TM1

or

QB1
RB1, RB2, RB3, RB4
WR1, WR2
TE1
PK1
TM1

or

QB2
RB1, RB2, RB3, RB5
WR1, WR2
TE1
PK1
TM1

and more . . .

The question - how many total different individual player combinations will there be for all the roster configurations with the starter limitations in place?  I'm thinking hundreds at least, possibly in the thousands?

Make sense?