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.)

## 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.

**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.

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**

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 272454480Phoenix Elves 20 272454480Cave Goblins 20 169033800Guild Dwarves 20 217543500Vanguard 20 215443080Fallen Kingdom 4 1095643824Jungle Elves 4 819732788Cloaks 20 217543500Benders 20 270354060Sand Goblins 20 217543500Deep Dwarves 20 215443080Shadow Elves 20 169033800Mountain Vargath 20 272454480Swamp Orcs 20 334866960Total645332

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.

## What about the Mercenaries?

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.

# Comments

## I LIKE TAU! commented:

I'll have fun trying out every FL deck, including using up to 6 mercs in it. What fun!*Posted on 2012-10-03.*

## I LIKE TAU! commented:

Well, the mercs are actually less than FK, now that the 5th part has come out.*Posted on 2012-10-07.*

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## tristak commented:

Gonna have fun trying out all 8 billion merc decks ;)Posted on 2012-10-03.