How Many Decks? Part 5: Mercenaries in the Mix

The Big Picture

Hopefully you will soon be receiving the newly published Reinforcement Packs for Summoner Wars, if you don't have them already.  One of them has the reinforcements for the Mercenary faction, so it is fitting that we are finally getting around to counting the number of decks we could possibly build when we add Mercenaries to the other factions.

We have waited until now to do this because it is more complicated than the previous calculations.  Adding the Mercenaries adds yet another level of variables: The number of Mercenaries is not fixed, but can be anywhere up to 6.  And, these Mercenaries can be any combination of Champions and Commons, subject to the limit of 3 Champions, or course.

Figure 1 shows all these different ways of adding Mercenaries to a deck.  For example, if a deck has 1 Mercenary card, it can be either a Champion or a Common; if it has 2 Mercenaries, it could have 2 Champions, 2 Commons, or 1 of each; and so on.  

The situation we have discussed so far -- having no Mercenaries at all -- is the one at the top.  It is truly the tip of the iceberg, because it is only 1 of 22 different possibilities that have to be calculated separately, since as far as I know there is no combinatorial shortcut to count all the decks at once, given the intersecting constraints about Champions, Commons, and Mercenaries.  

Simplifying the Picture

Fortunately, when we look at a problem we can often use common sense to make it easier to approach, even if there is no technical tool at hand. 

Figure 2. Sorting the Mercenary options by Champions and Commons included

Figure 2 shows the same deck options as in Figure 1, but sorted into rows and columns according to the number of Merc Commons and Champions they have.  Remembering that the calculations for the number of possible Champion combinations and Common combinations are independent of each other, we realize we don't have to do a separate calculation for each of the 22 types of decks, but just one for each of the columns and for each of the rows, and multiply them.  So if A, B, C and D are the number of combinations of Champions using 0, 1, 2 and 3 Mercs, and T-Z are the combinations for 0-6 Commons, we then calculate:

AT + AU + AV + AW + AX + AY + AZ

+ BT + BU + BV + BW + BX + BY

+ CT + CU + CV + CW + CX

+ DT + DU + DV + DW

There is one other possible shortcut: combine all the row and column entries, multiply them, and subtract the 6 types that are not included since they yield more than 6 Mercenaries: 

(A + B + C + D) * (T + U + V + W + X + Y + Z)

- (BZ + CY + CZ + DX + DY + DZ)

However, this is more of a shortcut when doing it by hand than when using a spreadsheet, like I did. 

One Detailed Example

Since this is a math demo, let's look at the calculations behind one of the possibilities, say when we include 2 Merc Champions and 4 Merc Commons in a Tundra Orc deck.

Figure 3. Building a deck using 6 Mercenaries -- 2 Champions and 4 Commons

There are 4 separate combinatorial calculations.  As described in Part 1 of this series, to choose 1 TO Champion out of 6 choices is: 

C( 6, 1 ) = 6! / 5!1! = 6 / 1 = 6

To choose 2 MR Champions out of 18 possibilities is:

C( 18, 2 ) = 18! / 16!2! = 18*17 / 2*1 = 153

As described in Part 2 and Part 3, to choose 4 MR Commons out of 13 kinds is:

C( 4+(13-1), (13-1) ) = C( 16, 12 ) = C( 16, 4 ) = 16*15*14*13 / 4*3*2*1 = 1820 

Finally the TO Commons have 10 slots left, since 4 slots are commited to the setup Commons.  We have 5 kinds to choose from, and we have to subtract combinations having more than 10 of either Fighters, Smashers or Shamans (note that with 10 slots we cannot choose too many Chargers or Shamans):

C( 10+(5-1), (5-1) ) = C( 14, 4 ) = 14*13*12*11 / 4*3*2*1 = 1001

C( 1+(5-1), (5-1) ) = C( 5, 4 ) = C( 5, 1 ) = 5 

C( 0+(5-1), (5-1) ) = C( 4, 4 ) = 1 

1001 - 5 - 1 - 1 = 994

Finally, multiplying them all together we get 

6 * 153 * 1820 * 994 = 1,660,735,440

That's quite a lot! 

Doing the Rest

Doing similar calculations for the other possibilities seen above, here are the results in a table formatted similar to Figure 2.  The numbers at the heads of the rows and columns are the number of combinations for that row or column, and the rest of the cells show those numbers multiplied:

This makes a grand total of 5,267,694,456!  That's in the same ballpark as the 7,868,950,752 that the Mercenary faction on its own has.  

Finally, here are the similar results for almost all the non-Mercenary factions.  The first column has no limits outside the normal deck-building rules, while the second column is limited to having one of each deck offered by PHG (see Part 4 for details):

The one faction missing above is the Filth.  They have their own deck-building issues different from any other faction, and they will be covered in the next article.  

Posted by Bryan Stout

Bryan Stout is a contributor to PlaidHatGames.com