Welcome to the third installment in this series that tries to count the number of distinct decks we can build in Summoner Wars.  In the previous parts we examined the number of combinations of Champions, and of Commons from a faction’s starting deck.  Now we’ll add in the Reinforcements!  (Including the new ones about to appear.) Figure 1. Tundra Orc Champions and Commons from the Starter Set and Reinforcement Pack

As before, we’ll use the Tundra Orcs as an example, and then calculate for the other factions after the principles are explained.

## Same Principles, Larger Numbers

If we count the number of all legal Tundra Orc decks using both the Starter Set and the Reinforcement Pack (and not using any Mercenaries), we already have all the mathematical tools at hand.  In Part 1 we saw that the number of ways to choose K objects out of N choices is

C( N, K ) = N! / (N-K!)K!

So we plug in the numbers to choose 3 Champions out of 6 possibilities and get C( 6, 3 ) = 6*5*4 / 3*2*1 = 20. Figure 2. Tundra Orc Champions and Commons from the Starter Set and Reinforcement Pack

In Part 2 we saw that the number of ways to choose K objects out of N different kinds of objects is

C( K+(N-1), N-1 )

Just as before, K=14, since we need 18 Common Units and 4 are mandated in the setup.  But now with the Reinforcements, N=5, not 3.  So, to start with, we have

C( 14+(5-1), 5-1 ) = C( 18, 4 ) = 18*17*16*15 / 4*3*2*1 = 3060

possible combinations of Commons, instead of 120. Figure 3. Choosing from 5 kinds of Tundra Orc

But now we need to subtract the illegal combinations, that is those that have more than 10 copies of any kind of Common.  To see the number of decks with too many Thwarters, we pick 11 Thwarters and see how many ways there are of filling the remaining 3 slots:

C( 3+(5-1), 5-1 ) = C( 7, 4 ) = 7*6*5*4 / 4*3*2*1 = 35 Figure 4. Choosing too many Thwarters, at least 11

The number of decks with too many Chargers is the same.  For both Smashers and Shamans we have 1 in the setup, so we need only 10 more to have too many, leaving 4 slots to fill:

C( 4+(5-1), 5-1 ) = C( 8, 4 ) = 8*7*6*5 / 4*3*2*1 = 70

We have 2 Fighters in the setup so we have 5 slots free to fill:

C( 5+(5-1), 5-1 ) = C( 9, 4 ) = 9*8*7*6 / 4*3*2*1 = 126

So the total number of legal combinations of Tundra Orc Commons is

3060 – 126 – 70 – 70 – 35 – 35 = 2724

And multiplying that times the number of Champion combinations yields

20 * 2724 = 54,480

That’s a lot of possible decks, isn’t it?  (And we’re not even including Mercenaries yet!)

Doing similar calculations for all the factions, we get:

```Faction      Champ Comb   Common Comb    Total Comb
Tundra Orcs          20          2724         54480
Phoenix Elves        20          2724         54480
Cave Goblins         20          1690         33800
Guild Dwarves        20          2175         43500
Vanguard             20          2154         43080
Fallen Kingdom        4         10956         43824
Jungle Elves          4          8197         32788
Cloaks               20          2175         43500
Benders              20          2703         54060
Sand Goblins         20          2175         43500
Deep Dwarves         20          2154         43080
Mountain Vargath     20          2724         54480
Swamp Orcs           20          3348         66960
Total                                        645332 ```

Some trends are apparent.  The factions with more Common units on the battlefield at the start of the game have fewer possible decks to build — which makes sense — compare the Swamp Orcs who start with 3, with the Phoenix Elves who start with 4, the Guild Dwarves with 5, and the Cave Goblins with 6.  Also note that the two factions with only 1 Champion in their Reinforcement Packs — the Fallen Kingdom and the Jungle Elves — have many fewer Champion combinations, but the extra kind of Common unit increases the number of Common combinations to almost make up for it.

You may have also noticed that two factions are missing in the list above: the Filth and Mercenaries, who have their own unique issues.  The Mercenaries by themselves are not that problematical, they just have many more units than other factions, so we have time to discuss them here.

There have been 18 Mercenary Champions published — 3 with the Merc Faction Deck, 11 with the Reinforcement Packs, and  4 promo cards — so the number of ways of choosing 3 of them is:

C( 18, 3 ) = 18*17*16 / 3*2*1 = 816

That's a lot more than 20, isn't it?  Trippling the number of Champions had a much bigger effect than we might have expected.  But we're only getting started: the Commons will show how very fast the factorial nature of combinations goes up.

We have 13 kinds of Commons, 3 in the Faction Deck and 10 from the Reinforcement Packs.  The Rallul starting setup uses 4 Commons, leaving 14 slots free, so

C( 14+(13-1), 13-1 ) = C( 26, 12 )

= 26*25*24*23*22*21*20*19*18*17*16*15 / 12*11*10*9*8*7*6*5*4*3*2*1

9,657,700

is the number of raw Common combinations, but first we subtract the illegal ones.  There are 2 Stone Golems in the setup, so adding 9 more leaves 5 slots:

C( 5+(13-1), 13-1 ) = C( 17, 12 ) = C( 17, 5 ) = 17*16*15*14*13 / 5*4*3*2*1 = 6188

There is 1 each of Rune Mages and Apprentice Mages, leaving 4 slots after adding 10 more:

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

All the 10 other kinds of Merc Commons are not in the setup, so we use up 11 slots for them, leaving 3:

C( 3+(13-1), 13-1 ) = C( 15, 12 ) = C( 15, 3 ) = 15*14*13 / 3*2*1 = 455

Giving us the actual Common combinations as

9,657,700 - 6188 - (2*1820) – (10*455) = 9,643,322

We multiply the Champion and Common results for a grand total of

816 * 9,643,322  = 7,868,950,752

There are close to 8 billion different Mercenary decks we can build.  The mind just boggles.