###########################################################
# Run the following code at https://sagecell.sagemath.org/
###########################################################

###########################################################
# PARAMETERS
###########################################################

N  = 113   # total regular cards
E  = 37    # cards with evo only
EH = 2     # cards with both evo and hero forms
H  = 2     # cards with hero only
C  = 8     # champions
K  = 8     # deck size

MAX_EVOS       = 2   # max evos in a deck
MAX_HERO_SLOTS = 2   # max hero-slot cards (heroes + champs) in a deck

TOWER_MULTIPLIER = 4 # tower troop factor


###########################################################
# T(e1, e2, h1, h2, h3)
#
# e1 = # of E cards used as evo
# e2 = # of EH cards used as evo
# h1 = # of EH cards used as hero
# h2 = # of H-only cards used as hero
# h3 = # of champions
#
# Constraints (enforced by loop ranges, but this is the logic):
# - e1 + e2 <= MAX_EVOS      (evo slots)
# - h1 + h2 + h3 <= MAX_HERO_SLOTS  (hero slots)
# - e2 + h1 <= EH            (can't use more EH cards than exist)
# - 0 <= e1 <= E, 0 <= e2 <= EH, etc.
###########################################################

def T(e1, e2, h1, h2, h3):

    # Choose specific cards for each role
    choose_E_evos      = binomial(E,  e1)
    choose_EH_evos     = binomial(EH, e2)
    choose_EH_heroes   = binomial(EH - e2, h1)
    choose_H_heroes    = binomial(H,  h2)
    choose_champions   = binomial(C,  h3)

    # Remaining slots in the 8-card deck must be regular-form cards
    special_count   = e1 + e2 + h1 + h2 + h3
    remaining_slots = K - special_count

    # Available regular-form cards:
    # total regular N minus those already used in a non-regular form
    regular_available = N - e1 - e2 - h1 - h2

    choose_regulars = binomial(regular_available, remaining_slots)

    return (choose_E_evos *
            choose_EH_evos *
            choose_EH_heroes *
            choose_H_heroes *
            choose_champions *
            choose_regulars)


###########################################################
# SUM OVER ALL VALID (e1, e2, h1, h2, h3)
#
# Loop bounds:
#
# - e1: 0 .. min(E, MAX_EVOS)
# - e2: 0 .. min(EH, MAX_EVOS - e1)
# - h1: 0 .. min(EH - e2, MAX_HERO_SLOTS)
# - h2: 0 .. min(H, MAX_HERO_SLOTS - h1)
# - h3: 0 .. min(C, MAX_HERO_SLOTS - h1 - h2)
#
# These ensure:
#   e1 + e2 <= MAX_EVOS
#   h1 + h2 + h3 <= MAX_HERO_SLOTS
#   e2 + h1 <= EH       (since h1 <= EH - e2)
###########################################################

total = 0

for e1 in range(0, min(E, MAX_EVOS) + 1):
    for e2 in range(0, min(EH, MAX_EVOS - e1) + 1):
        # EH cards left: EH - e2
        max_h1 = min(EH - e2, MAX_HERO_SLOTS)
        for h1 in range(0, max_h1 + 1):
            max_h2 = min(H, MAX_HERO_SLOTS - h1)
            for h2 in range(0, max_h2 + 1):
                max_h3 = min(C, MAX_HERO_SLOTS - h1 - h2)
                for h3 in range(0, max_h3 + 1):
                    total += T(e1, e2, h1, h2, h3)

result = TOWER_MULTIPLIER * total
result