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A network flow problem need a elegant solution(python)

I am using a networkx package from python and I have a dataframe
(Sample dataframe)
from to count
v0 v1 0.1
v0 v2 0.15
v0 v3 0.15
v0 v4 0.25
v0 v5 0.15
and so on..
Sample picture(weighted direct graph)
That is my dataframe.
{'grad': {0: 'CUHK', 1: 'CUHK', 2: 'CUHK', 3: 'CUHK', 4: 'CUHK', 5: 'CityU', 6: 'CityU', 7: 'CityU', 8: 'CityU', 9: 'HKU', 10: 'HKU', 11: 'HKU', 12: 'HKUST', 13: 'HKUST', 14: 'HKUST', 15: 'HKUST', 16: 'HKUST', 17: 'HKUST', 18: 'Low Frequency', 19: 'Low Frequency', 20: 'Low Frequency', 21: 'Low Frequency', 22: 'Low Frequency', 23: 'Low Frequency', 24: 'PolyU', 25: 'PolyU', 26: 'PolyU', 27: 'PolyU'}, 'to': {0: 'CUHK', 1: 'CityU', 2: 'HKU', 3: 'LingU', 4: 'PolyU', 5: 'CityU', 6: 'HKU', 7: 'LingU', 8: 'PolyU', 9: 'CityU', 10: 'HKU', 11: 'PolyU', 12: 'CUHK', 13: 'CityU', 14: 'HKU', 15: 'HKUST', 16: 'LingU', 17: 'PolyU', 18: 'CUHK', 19: 'CityU', 20: 'HKU', 21: 'HKUST', 22: 'LingU', 23: 'PolyU', 24: 'CityU', 25: 'HKU', 26: 'LingU', 27: 'PolyU'}, 'count': {0: 9, 1: 5, 2: 3, 3: 2, 4: 3, 5: 3, 6: 2, 7: 2, 8: 3, 9: 3, 10: 9, 11: 4, 12: 2, 13: 1, 14: 2, 15: 1, 16: 4, 17: 4, 18: 49, 19: 34, 20: 29, 21: 34, 22: 3, 23: 36, 24: 1, 25: 1, 26: 1, 27: 11}}
The principle of ranking is when Vx -> Vy is bigger than Vy -> Vx, Vx has a higher rank than Vy.
e.g. V0 -> V5 = 0.2 and V5 -> V0 = 0.5 so, V5 have a higher rank
Now I am using the brute force method, which loops and checks all the relationships. When the condition is met, I change their order in a new list. -> {V0,V1,V2,V3,V4,V5,V6,V7}
I want an elegant solution to rank these nodes. Maybe I can get some partial orders like V5>V0 and V0>V1 and use them to form a global order V5>V0>V1, but I don't know how to achieve it. Is there any method better than brute force? Is this related to any famous problem?

One way of doing this would be the following:
import networkx as nx
import pandas as pd
data = {'grad': {0: 'CUHK', 1: 'CUHK', 2: 'CUHK', 3: 'CUHK', 4: 'CUHK', 5: 'CityU', 6: 'CityU', 7: 'CityU', 8: 'CityU', 9: 'HKU', 10: 'HKU', 11: 'HKU', 12: 'HKUST', 13: 'HKUST', 14: 'HKUST', 15: 'HKUST', 16: 'HKUST', 17: 'HKUST', 18: 'Low Frequency', 19: 'Low Frequency', 20: 'Low Frequency', 21: 'Low Frequency', 22: 'Low Frequency', 23: 'Low Frequency', 24: 'PolyU', 25: 'PolyU', 26: 'PolyU', 27: 'PolyU'},
'to': {0: 'CUHK', 1: 'CityU', 2: 'HKU', 3: 'LingU', 4: 'PolyU', 5: 'CityU', 6: 'HKU', 7: 'LingU', 8: 'PolyU', 9: 'CityU', 10: 'HKU', 11: 'PolyU', 12: 'CUHK', 13: 'CityU', 14: 'HKU', 15: 'HKUST', 16: 'LingU', 17: 'PolyU', 18: 'CUHK', 19: 'CityU', 20: 'HKU', 21: 'HKUST', 22: 'LingU', 23: 'PolyU', 24: 'CityU', 25: 'HKU', 26: 'LingU', 27: 'PolyU'},
'count': {0: 9, 1: 5, 2: 3, 3: 2, 4: 3, 5: 3, 6: 2, 7: 2, 8: 3, 9: 3, 10: 9, 11: 4, 12: 2, 13: 1, 14: 2, 15: 1, 16: 4, 17: 4, 18: 49, 19: 34, 20: 29, 21: 34, 22: 3, 23: 36, 24: 1, 25: 1, 26: 1, 27: 11}}
df = pd.DataFrame(data)
G = nx.from_pandas_edgelist(df, 'grad', 'to', edge_attr='count', create_using=nx.DiGraph())
pagerank = nx.pagerank(G, weight='count')
sorted_pagerank = sorted(pagerank.items(), key=lambda x: x[1], reverse=True)
This returns a list of tuples with the node and its PageRank score, sorted in descending order of the PageRank score.
[('PolyU', 0.4113039270586079),
('HKU', 0.1945013448661985),
('CityU', 0.14888513201115303),
('LingU', 0.09978025157613143),
('CUHK', 0.07069262490080512),
('HKUST', 0.041291981078138223),
('Low Frequency', 0.03354473850896578)]
If you want this with the graph:
import matplotlib.pyplot as plt
import networkx as nx
G = nx.from_pandas_edgelist(df, 'grad', 'to', edge_attr='count', create_using=nx.DiGraph())
pagerank = nx.pagerank(G, weight='count')
sorted_pagerank = sorted(pagerank.items(), key=lambda x: x[1], reverse=True)
pos = nx.spring_layout(G)
nx.draw(G, pos, with_labels=True, node_color='skyblue', node_size=[v * 100 for v in pagerank.values()])
labels = nx.get_edge_attributes(G,'count')
nx.draw_networkx_edge_labels(G, pos, edge_labels=labels)
plt.show()

python title disappears when trying to align it to the left in seaborn

Here is the dataframe I'm working with in python. I'm including the dataframe here with this line of code:
print(mtcars.to_dict())
{'Unnamed: 0': {0: 'Mazda RX4', 1: 'Mazda RX4 Wag', 2: 'Datsun 710', 3: 'Hornet 4 Drive', 4: 'Hornet Sportabout', 5: 'Valiant', 6: 'Duster 360', 7: 'Merc 240D', 8: 'Merc 230', 9: 'Merc 280', 10: 'Merc 280C', 11: 'Merc 450SE', 12: 'Merc 450SL', 13: 'Merc 450SLC', 14: 'Cadillac Fleetwood', 15: 'Lincoln Continental', 16: 'Chrysler Imperial', 17: 'Fiat 128', 18: 'Honda Civic', 19: 'Toyota Corolla', 20: 'Toyota Corona', 21: 'Dodge Challenger', 22: 'AMC Javelin', 23: 'Camaro Z28', 24: 'Pontiac Firebird', 25: 'Fiat X1-9', 26: 'Porsche 914-2', 27: 'Lotus Europa', 28: 'Ford Pantera L', 29: 'Ferrari Dino', 30: 'Maserati Bora', 31: 'Volvo 142E'}, 'mpg': {0: 21.0, 1: 21.0, 2: 22.8, 3: 21.4, 4: 18.7, 5: 18.1, 6: 14.3, 7: 24.4, 8: 22.8, 9: 19.2, 10: 17.8, 11: 16.4, 12: 17.3, 13: 15.2, 14: 10.4, 15: 10.4, 16: 14.7, 17: 32.4, 18: 30.4, 19: 33.9, 20: 21.5, 21: 15.5, 22: 15.2, 23: 13.3, 24: 19.2, 25: 27.3, 26: 26.0, 27: 30.4, 28: 15.8, 29: 19.7, 30: 15.0, 31: 21.4}, 'cyl': {0: 6, 1: 6, 2: 4, 3: 6, 4: 8, 5: 6, 6: 8, 7: 4, 8: 4, 9: 6, 10: 6, 11: 8, 12: 8, 13: 8, 14: 8, 15: 8, 16: 8, 17: 4, 18: 4, 19: 4, 20: 4, 21: 8, 22: 8, 23: 8, 24: 8, 25: 4, 26: 4, 27: 4, 28: 8, 29: 6, 30: 8, 31: 4}, 'disp': {0: 160.0, 1: 160.0, 2: 108.0, 3: 258.0, 4: 360.0, 5: 225.0, 6: 360.0, 7: 146.7, 8: 140.8, 9: 167.6, 10: 167.6, 11: 275.8, 12: 275.8, 13: 275.8, 14: 472.0, 15: 460.0, 16: 440.0, 17: 78.7, 18: 75.7, 19: 71.1, 20: 120.1, 21: 318.0, 22: 304.0, 23: 350.0, 24: 400.0, 25: 79.0, 26: 120.3, 27: 95.1, 28: 351.0, 29: 145.0, 30: 301.0, 31: 121.0}, 'hp': {0: 110, 1: 110, 2: 93, 3: 110, 4: 175, 5: 105, 6: 245, 7: 62, 8: 95, 9: 123, 10: 123, 11: 180, 12: 180, 13: 180, 14: 205, 15: 215, 16: 230, 17: 66, 18: 52, 19: 65, 20: 97, 21: 150, 22: 150, 23: 245, 24: 175, 25: 66, 26: 91, 27: 113, 28: 264, 29: 175, 30: 335, 31: 109}, 'drat': {0: 3.9, 1: 3.9, 2: 3.85, 3: 3.08, 4: 3.15, 5: 2.76, 6: 3.21, 7: 3.69, 8: 3.92, 9: 3.92, 10: 3.92, 11: 3.07, 12: 3.07, 13: 3.07, 14: 2.93, 15: 3.0, 16: 3.23, 17: 4.08, 18: 4.93, 19: 4.22, 20: 3.7, 21: 2.76, 22: 3.15, 23: 3.73, 24: 3.08, 25: 4.08, 26: 4.43, 27: 3.77, 28: 4.22, 29: 3.62, 30: 3.54, 31: 4.11}, 'wt': {0: 2.62, 1: 2.875, 2: 2.32, 3: 3.215, 4: 3.44, 5: 3.46, 6: 3.57, 7: 3.19, 8: 3.15, 9: 3.44, 10: 3.44, 11: 4.07, 12: 3.73, 13: 3.78, 14: 5.25, 15: 5.424, 16: 5.345, 17: 2.2, 18: 1.615, 19: 1.835, 20: 2.465, 21: 3.52, 22: 3.435, 23: 3.84, 24: 3.845, 25: 1.935, 26: 2.14, 27: 1.513, 28: 3.17, 29: 2.77, 30: 3.57, 31: 2.78}, 'qsec': {0: 16.46, 1: 17.02, 2: 18.61, 3: 19.44, 4: 17.02, 5: 20.22, 6: 15.84, 7: 20.0, 8: 22.9, 9: 18.3, 10: 18.9, 11: 17.4, 12: 17.6, 13: 18.0, 14: 17.98, 15: 17.82, 16: 17.42, 17: 19.47, 18: 18.52, 19: 19.9, 20: 20.01, 21: 16.87, 22: 17.3, 23: 15.41, 24: 17.05, 25: 18.9, 26: 16.7, 27: 16.9, 28: 14.5, 29: 15.5, 30: 14.6, 31: 18.6}, 'vs': {0: 0, 1: 0, 2: 1, 3: 1, 4: 0, 5: 1, 6: 0, 7: 1, 8: 1, 9: 1, 10: 1, 11: 0, 12: 0, 13: 0, 14: 0, 15: 0, 16: 0, 17: 1, 18: 1, 19: 1, 20: 1, 21: 0, 22: 0, 23: 0, 24: 0, 25: 1, 26: 0, 27: 1, 28: 0, 29: 0, 30: 0, 31: 1}, 'am': {0: 1, 1: 1, 2: 1, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0, 10: 0, 11: 0, 12: 0, 13: 0, 14: 0, 15: 0, 16: 0, 17: 1, 18: 1, 19: 1, 20: 0, 21: 0, 22: 0, 23: 0, 24: 0, 25: 1, 26: 1, 27: 1, 28: 1, 29: 1, 30: 1, 31: 1}, 'gear': {0: 4, 1: 4, 2: 4, 3: 3, 4: 3, 5: 3, 6: 3, 7: 4, 8: 4, 9: 4, 10: 4, 11: 3, 12: 3, 13: 3, 14: 3, 15: 3, 16: 3, 17: 4, 18: 4, 19: 4, 20: 3, 21: 3, 22: 3, 23: 3, 24: 3, 25: 4, 26: 5, 27: 5, 28: 5, 29: 5, 30: 5, 31: 4}, 'carb': {0: 4, 1: 4, 2: 1, 3: 1, 4: 2, 5: 1, 6: 4, 7: 2, 8: 2, 9: 4, 10: 4, 11: 3, 12: 3, 13: 3, 14: 4, 15: 4, 16: 4, 17: 1, 18: 2, 19: 1, 20: 1, 21: 2, 22: 2, 23: 4, 24: 2, 25: 1, 26: 2, 27: 2, 28: 4, 29: 6, 30: 8, 31: 2}}
This SO post was helpful in learning how to print the python dataframe like R does with the dput() function.
Now I import seaborn and create a histogram.
import seaborn as seaborn
seaborn.histplot(data=mtcars, x="mpg", bins = 30)
plt.suptitle("Mtcars", loc = 'left')
plt.title("histogram", loc = 'left')
plt.show()
This doesn't work as the title disappears.
So I clear out whatever is happening with the graphs and try again.
plt.figure().clear()
plt.close()
plt.cla()
plt.clf()
seaborn.histplot(data=mtcars, x="mpg", bins = 30)
plt.suptitle("Mtcars", horizontalalignment = 'left')
plt.title("histogram", loc = 'left')
plt.show()
But this doesn't work either. This time, the title is there but the alignment is wrong.
I'd like to put both the title and the subtitle on the left side.

Python trying to create a graph but it's blank

Here is the dataframe that I'm working with in python.
{'Unnamed: 0': {0: 1, 1: 2, 2: 3, 3: 4, 4: 5, 5: 6, 6: 7, 7: 8, 8: 9, 9: 10, 10: 11, 11: 12, 12: 13, 13: 14, 14: 15, 15: 16, 16: 17, 17: 18, 18: 19, 19: 20, 20: 21, 21: 22, 22: 23, 23: 24, 24: 25, 25: 26, 26: 27, 27: 28, 28: 29, 29: 30, 30: 31, 31: 32}, 'car': {0: 'Mazda RX4', 1: 'Mazda RX4 Wag', 2: 'Datsun 710', 3: 'Hornet 4 Drive', 4: 'Hornet Sportabout', 5: 'Valiant', 6: 'Duster 360', 7: 'Merc 240D', 8: 'Merc 230', 9: 'Merc 280', 10: 'Merc 280C', 11: 'Merc 450SE', 12: 'Merc 450SL', 13: 'Merc 450SLC', 14: 'Cadillac Fleetwood', 15: 'Lincoln Continental', 16: 'Chrysler Imperial', 17: 'Fiat 128', 18: 'Honda Civic', 19: 'Toyota Corolla', 20: 'Toyota Corona', 21: 'Dodge Challenger', 22: 'AMC Javelin', 23: 'Camaro Z28', 24: 'Pontiac Firebird', 25: 'Fiat X1-9', 26: 'Porsche 914-2', 27: 'Lotus Europa', 28: 'Ford Pantera L', 29: 'Ferrari Dino', 30: 'Maserati Bora', 31: 'Volvo 142E'}, 'mpg': {0: 21.0, 1: 21.0, 2: 22.8, 3: 21.4, 4: 18.7, 5: 18.1, 6: 14.3, 7: 24.4, 8: 22.8, 9: 19.2, 10: 17.8, 11: 16.4, 12: 17.3, 13: 15.2, 14: 10.4, 15: 10.4, 16: 14.7, 17: 32.4, 18: 30.4, 19: 33.9, 20: 21.5, 21: 15.5, 22: 15.2, 23: 13.3, 24: 19.2, 25: 27.3, 26: 26.0, 27: 30.4, 28: 15.8, 29: 19.7, 30: 15.0, 31: 21.4}, 'cyl': {0: 6, 1: 6, 2: 4, 3: 6, 4: 8, 5: 6, 6: 8, 7: 4, 8: 4, 9: 6, 10: 6, 11: 8, 12: 8, 13: 8, 14: 8, 15: 8, 16: 8, 17: 4, 18: 4, 19: 4, 20: 4, 21: 8, 22: 8, 23: 8, 24: 8, 25: 4, 26: 4, 27: 4, 28: 8, 29: 6, 30: 8, 31: 4}, 'disp': {0: 160.0, 1: 160.0, 2: 108.0, 3: 258.0, 4: 360.0, 5: 225.0, 6: 360.0, 7: 146.7, 8: 140.8, 9: 167.6, 10: 167.6, 11: 275.8, 12: 275.8, 13: 275.8, 14: 472.0, 15: 460.0, 16: 440.0, 17: 78.7, 18: 75.7, 19: 71.1, 20: 120.1, 21: 318.0, 22: 304.0, 23: 350.0, 24: 400.0, 25: 79.0, 26: 120.3, 27: 95.1, 28: 351.0, 29: 145.0, 30: 301.0, 31: 121.0}, 'hp': {0: 110, 1: 110, 2: 93, 3: 110, 4: 175, 5: 105, 6: 245, 7: 62, 8: 95, 9: 123, 10: 123, 11: 180, 12: 180, 13: 180, 14: 205, 15: 215, 16: 230, 17: 66, 18: 52, 19: 65, 20: 97, 21: 150, 22: 150, 23: 245, 24: 175, 25: 66, 26: 91, 27: 113, 28: 264, 29: 175, 30: 335, 31: 109}, 'drat': {0: 3.9, 1: 3.9, 2: 3.85, 3: 3.08, 4: 3.15, 5: 2.76, 6: 3.21, 7: 3.69, 8: 3.92, 9: 3.92, 10: 3.92, 11: 3.07, 12: 3.07, 13: 3.07, 14: 2.93, 15: 3.0, 16: 3.23, 17: 4.08, 18: 4.93, 19: 4.22, 20: 3.7, 21: 2.76, 22: 3.15, 23: 3.73, 24: 3.08, 25: 4.08, 26: 4.43, 27: 3.77, 28: 4.22, 29: 3.62, 30: 3.54, 31: 4.11}, 'wt': {0: 2.62, 1: 2.875, 2: 2.32, 3: 3.215, 4: 3.44, 5: 3.46, 6: 3.57, 7: 3.19, 8: 3.15, 9: 3.44, 10: 3.44, 11: 4.07, 12: 3.73, 13: 3.78, 14: 5.25, 15: 5.424, 16: 5.345, 17: 2.2, 18: 1.615, 19: 1.835, 20: 2.465, 21: 3.52, 22: 3.435, 23: 3.84, 24: 3.845, 25: 1.935, 26: 2.14, 27: 1.513, 28: 3.17, 29: 2.77, 30: 3.57, 31: 2.78}, 'qsec': {0: 16.46, 1: 17.02, 2: 18.61, 3: 19.44, 4: 17.02, 5: 20.22, 6: 15.84, 7: 20.0, 8: 22.9, 9: 18.3, 10: 18.9, 11: 17.4, 12: 17.6, 13: 18.0, 14: 17.98, 15: 17.82, 16: 17.42, 17: 19.47, 18: 18.52, 19: 19.9, 20: 20.01, 21: 16.87, 22: 17.3, 23: 15.41, 24: 17.05, 25: 18.9, 26: 16.7, 27: 16.9, 28: 14.5, 29: 15.5, 30: 14.6, 31: 18.6}, 'vs': {0: 0, 1: 0, 2: 1, 3: 1, 4: 0, 5: 1, 6: 0, 7: 1, 8: 1, 9: 1, 10: 1, 11: 0, 12: 0, 13: 0, 14: 0, 15: 0, 16: 0, 17: 1, 18: 1, 19: 1, 20: 1, 21: 0, 22: 0, 23: 0, 24: 0, 25: 1, 26: 0, 27: 1, 28: 0, 29: 0, 30: 0, 31: 1}, 'am': {0: 1, 1: 1, 2: 1, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0, 10: 0, 11: 0, 12: 0, 13: 0, 14: 0, 15: 0, 16: 0, 17: 1, 18: 1, 19: 1, 20: 0, 21: 0, 22: 0, 23: 0, 24: 0, 25: 1, 26: 1, 27: 1, 28: 1, 29: 1, 30: 1, 31: 1}, 'gear': {0: 4, 1: 4, 2: 4, 3: 3, 4: 3, 5: 3, 6: 3, 7: 4, 8: 4, 9: 4, 10: 4, 11: 3, 12: 3, 13: 3, 14: 3, 15: 3, 16: 3, 17: 4, 18: 4, 19: 4, 20: 3, 21: 3, 22: 3, 23: 3, 24: 3, 25: 4, 26: 5, 27: 5, 28: 5, 29: 5, 30: 5, 31: 4}, 'carb': {0: 4, 1: 4, 2: 1, 3: 1, 4: 2, 5: 1, 6: 4, 7: 2, 8: 2, 9: 4, 10: 4, 11: 3, 12: 3, 13: 3, 14: 4, 15: 4, 16: 4, 17: 1, 18: 2, 19: 1, 20: 1, 21: 2, 22: 2, 23: 4, 24: 2, 25: 1, 26: 2, 27: 2, 28: 4, 29: 6, 30: 8, 31: 2}}
Here is the code that I'm using. The subplot part I got off a datacamp module.
fig, ax = plt.subplot()
plt.show()
But when I go to plot the mtcars dataset, one variable against the other, I get a blank canvas. Why is that? I don't see how the code is different than what I am looking at on DataCamp.
ax.plot(mtcars['cyl'], mtcars['mpg'])
plt.show()
The answer from below is helpful and gets me closer to a solution but it is giving me lines instead of a scatterplot?
fig, ax = plt.subplot()
plt.show()

import matplotlib.pyplot as plt
plt.plot(df['cyl'], df['mpg'])
plt.show()
or:
ax = plt.subplot(2, 1, 1)
ax.plot(df['cyl'], df['mpg'])
plt.show()

Is there a formulaic approach to find the frequency of the sum of combinations?

I have 5 strawberries, 2 lemons, and a banana. For each possible combination of these (including selecting 0), there is a total number of objects. I ultimately want a list of the frequencies at which these sums appear.
[1 strawberry, 0 lemons, 0 bananas] = 1 objects
[2 strawberries, 0 lemons, 1 banana] = 3 objects
[0 strawberries, 1 lemon, 0 bananas] = 1 objects
[2 strawberries, 1 lemon, 0 bananas] = 3 objects
[3 strawberries, 0 lemons, 0 bananas] = 3 objects
For just the above selection of 5 combinations, "1" has a frequency of 2 and "3" has a frequency of 3.
Obviously there are far more possible combinations, each changing the frequency result. Is there a formulaic way to approach the problem to find the frequencies for an entire set of combinations?
Currently, I've set up a brute-force function in Python.
special_cards = {
'A':7, 'B':1, 'C':1, 'D':1, 'E':1, 'F':1, 'G':1, 'H':1, 'I':1, 'J':1, 'K':1, 'L':1,
'M':1, 'N':1, 'O':1, 'P':1, 'Q':1, 'R':1, 'S':1, 'T':1, 'U':1, 'V':1, 'W':1, 'X':1,
'Y':1, 'Z':1, 'AA':1, 'AB':1, 'AC':1, 'AD':1, 'AE':1, 'AF':1, 'AG':1, 'AH':1, 'AI':1, 'AJ':1,
'AK':1, 'AL':1, 'AM':1, 'AN':1, 'AO':1, 'AP':1, 'AQ':1, 'AR':1, 'AS':1, 'AT':1, 'AU':1, 'AV':1,
'AW':1, 'AX':1, 'AY':1
}
def _calc_dis_specials(special_cards):
"""Calculate the total combinations when special cards are factored in"""
# Create an iterator for special card combinations.
special_paths = _gen_dis_special_list(special_cards)
freq = {}
path_count = 0
for o_path in special_paths: # Loop through the iterator
path_count += 1 # Keep track of how many combinations we've evaluated thus far.
try: # I've been told I can use a collections.counter() object instead of try/except.
path_sum = sum(o_path) # Sum the path (counting objects)
new_count = freq[path_sum] + 1 # Try to increment the count for our sum.
freq.update({path_sum: new_count})
except KeyError:
freq.update({path_sum: 1})
print(f"{path_count:,}\n{freq}")
print(f"{path_count:,}\n{freq}")
# Do things with results yadda yadda
def _gen_dis_special_list(special_cards):
"""Generates an iterator for all combinations for special cards"""
product_args = []
for value in special_cards.values(): # A card's "value" is the maximum number that can be in a deck.
product_args.append(range(value+1)) # Populates product_args with lists of each card's possible count.
result = itertools.product(*product_args)
return result
However, for large numbers of object pools (50+) the factorial just gets out of hand. Billions upon billions of combinations. I need a formulaic approach.
Looking at some output, I notice a couple of things:
1
{0: 1}
2
{0: 1, 1: 1}
4
{0: 1, 1: 2, 2: 1}
8
{0: 1, 1: 3, 2: 3, 3: 1}
16
{0: 1, 1: 4, 2: 6, 3: 4, 4: 1}
32
{0: 1, 1: 5, 2: 10, 3: 10, 4: 5, 5: 1}
64
{0: 1, 1: 6, 2: 15, 3: 20, 4: 15, 5: 6, 6: 1}
128
{0: 1, 1: 7, 2: 21, 3: 35, 4: 35, 5: 21, 6: 7, 7: 1}
256
{0: 1, 1: 8, 2: 28, 3: 56, 4: 70, 5: 56, 6: 28, 7: 8, 8: 1}
512
{0: 1, 1: 9, 2: 36, 3: 84, 4: 126, 5: 126, 6: 84, 7: 36, 8: 9, 9: 1}
1,024
{0: 1, 1: 10, 2: 45, 3: 120, 4: 210, 5: 252, 6: 210, 7: 120, 8: 45, 9: 10, 10: 1}
2,048
{0: 1, 1: 11, 2: 55, 3: 165, 4: 330, 5: 462, 6: 462, 7: 330, 8: 165, 9: 55, 10: 11, 11: 1}
4,096
{0: 1, 1: 12, 2: 66, 3: 220, 4: 495, 5: 792, 6: 924, 7: 792, 8: 495, 9: 220, 10: 66, 11: 12, 12: 1}
8,192
{0: 1, 1: 13, 2: 78, 3: 286, 4: 715, 5: 1287, 6: 1716, 7: 1716, 8: 1287, 9: 715, 10: 286, 11: 78, 12: 13, 13: 1}
16,384
{0: 1, 1: 14, 2: 91, 3: 364, 4: 1001, 5: 2002, 6: 3003, 7: 3432, 8: 3003, 9: 2002, 10: 1001, 11: 364, 12: 91, 13: 14, 14: 1}
32,768
{0: 1, 1: 15, 2: 105, 3: 455, 4: 1365, 5: 3003, 6: 5005, 7: 6435, 8: 6435, 9: 5005, 10: 3003, 11: 1365, 12: 455, 13: 105, 14: 15, 15: 1}
65,536
{0: 1, 1: 16, 2: 120, 3: 560, 4: 1820, 5: 4368, 6: 8008, 7: 11440, 8: 12870, 9: 11440, 10: 8008, 11: 4368, 12: 1820, 13: 560, 14: 120, 15: 16, 16: 1}
131,072
{0: 1, 1: 17, 2: 136, 3: 680, 4: 2380, 5: 6188, 6: 12376, 7: 19448, 8: 24310, 9: 24310, 10: 19448, 11: 12376, 12: 6188, 13: 2380, 14: 680, 15: 136, 16: 17, 17: 1}
262,144
{0: 1, 1: 18, 2: 153, 3: 816, 4: 3060, 5: 8568, 6: 18564, 7: 31824, 8: 43758, 9: 48620, 10: 43758, 11: 31824, 12: 18564, 13: 8568, 14: 3060, 15: 816, 16: 153, 17: 18, 18: 1}
524,288
{0: 1, 1: 19, 2: 171, 3: 969, 4: 3876, 5: 11628, 6: 27132, 7: 50388, 8: 75582, 9: 92378, 10: 92378, 11: 75582, 12: 50388, 13: 27132, 14: 11628, 15: 3876, 16: 969, 17: 171, 18: 19, 19: 1}
1,048,576
{0: 1, 1: 20, 2: 190, 3: 1140, 4: 4845, 5: 15504, 6: 38760, 7: 77520, 8: 125970, 9: 167960, 10: 184756, 11: 167960, 12: 125970, 13: 77520, 14: 38760, 15: 15504, 16: 4845, 17: 1140, 18: 190, 19: 20, 20: 1}
2,097,152
{0: 1, 1: 21, 2: 210, 3: 1330, 4: 5985, 5: 20349, 6: 54264, 7: 116280, 8: 203490, 9: 293930, 10: 352716, 11: 352716, 12: 293930, 13: 203490, 14: 116280, 15: 54264, 16: 20349, 17: 5985, 18: 1330, 19: 210, 20: 21, 21: 1}
4,194,304
{0: 1, 1: 22, 2: 231, 3: 1540, 4: 7315, 5: 26334, 6: 74613, 7: 170544, 8: 319770, 9: 497420, 10: 646646, 11: 705432, 12: 646646, 13: 497420, 14: 319770, 15: 170544, 16: 74613, 17: 26334, 18: 7315, 19: 1540, 20: 231, 21: 22, 22: 1}
8,388,608
{0: 1, 1: 23, 2: 253, 3: 1771, 4: 8855, 5: 33649, 6: 100947, 7: 245157, 8: 490314, 9: 817190, 10: 1144066, 11: 1352078, 12: 1352078, 13: 1144066, 14: 817190, 15: 490314, 16: 245157, 17: 100947, 18: 33649, 19: 8855, 20: 1771, 21: 253, 22: 23, 23: 1}
16,777,216
{0: 1, 1: 24, 2: 276, 3: 2024, 4: 10626, 5: 42504, 6: 134596, 7: 346104, 8: 735471, 9: 1307504, 10: 1961256, 11: 2496144, 12: 2704156, 13: 2496144, 14: 1961256, 15: 1307504, 16: 735471, 17: 346104, 18: 134596, 19: 42504, 20: 10626, 21: 2024, 22: 276, 23: 24, 24: 1}
33,554,432
{0: 1, 1: 25, 2: 300, 3: 2300, 4: 12650, 5: 53130, 6: 177100, 7: 480700, 8: 1081575, 9: 2042975, 10: 3268760, 11: 4457400, 12: 5200300, 13: 5200300, 14: 4457400, 15: 3268760, 16: 2042975, 17: 1081575, 18: 480700, 19: 177100, 20: 53130, 21: 12650, 22: 2300, 23: 300, 24: 25, 25: 1}
67,108,864
{0: 1, 1: 26, 2: 325, 3: 2600, 4: 14950, 5: 65780, 6: 230230, 7: 657800, 8: 1562275, 9: 3124550, 10: 5311735, 11: 7726160, 12: 9657700, 13: 10400600, 14: 9657700, 15: 7726160, 16: 5311735, 17: 3124550, 18: 1562275, 19: 657800, 20: 230230, 21: 65780, 22: 14950, 23: 2600, 24: 325, 25: 26, 26: 1}
134,217,728
{0: 1, 1: 27, 2: 351, 3: 2925, 4: 17550, 5: 80730, 6: 296010, 7: 888030, 8: 2220075, 9: 4686825, 10: 8436285, 11: 13037895, 12: 17383860, 13: 20058300, 14: 20058300, 15: 17383860, 16: 13037895, 17: 8436285, 18: 4686825, 19: 2220075, 20: 888030, 21: 296010, 22: 80730, 23: 17550, 24: 2925, 25: 351, 26: 27, 27: 1}
268,435,456
{0: 1, 1: 28, 2: 378, 3: 3276, 4: 20475, 5: 98280, 6: 376740, 7: 1184040, 8: 3108105, 9: 6906900, 10: 13123110, 11: 21474180, 12: 30421755, 13: 37442160, 14: 40116600, 15: 37442160, 16: 30421755, 17: 21474180, 18: 13123110, 19: 6906900, 20: 3108105, 21: 1184040, 22: 376740, 23: 98280, 24: 20475, 25: 3276, 26: 378, 27: 28, 28: 1}
536,870,912
{0: 1, 1: 29, 2: 406, 3: 3654, 4: 23751, 5: 118755, 6: 475020, 7: 1560780, 8: 4292145, 9: 10015005, 10: 20030010, 11: 34597290, 12: 51895935, 13: 67863915, 14: 77558760, 15: 77558760, 16: 67863915, 17: 51895935, 18: 34597290, 19: 20030010, 20: 10015005, 21: 4292145, 22: 1560780, 23: 475020, 24: 118755, 25: 23751, 26: 3654, 27: 406, 28: 29, 29: 1}
1,073,741,824
{0: 1, 1: 30, 2: 435, 3: 4060, 4: 27405, 5: 142506, 6: 593775, 7: 2035800, 8: 5852925, 9: 14307150, 10: 30045015, 11: 54627300, 12: 86493225, 13: 119759850, 14: 145422675, 15: 155117520, 16: 145422675, 17: 119759850, 18: 86493225, 19: 54627300, 20: 30045015, 21: 14307150, 22: 5852925, 23: 2035800, 24: 593775, 25: 142506, 26: 27405, 27: 4060, 28: 435, 29: 30, 30: 1}
Note that I'm only printing when a new key (sum) is found.
I notice that
a new sum is found only on powers of 2 and
the results are symmetrical.
This hints to me that there's a formulaic approach that could work.
Any ideas on how to proceed?

Good news; there is a formula for this, and I'll explain the path there in case there is any confusion.
Let's look at your initial example: 5 strawberries (S), 2 lemons (L), and a banana (B). Let's lay out all of the fruits:
S S S S S L L B
We can actually rephrase the question now, because the number of times that 3, for example, will be the total number is the number of different ways you can pick 3 of the fruits from this list.
In statistics, the choose function (a.k.a nCk), answers just this question: how many ways are there to select a group of k items from a group of n items. This is computed as n!/((n-k)!*k!), where "!" is the factorial, a number multiplied by all numbers less than itself. As such the frequency of 3s would be (the number of fruits) "choose" (the total in question), or 8 choose 3. This is 8!/(5!*3!) = 56.

TypeError: unsupported operand type(s) for &: 'str' and 'bool'

All,
I have below Pandas dataframe, and I am trying to filter my dataframe such that my output displays country name along with the year 1989 column whose number is >1000000.For this I am using below code, but it is returning me below error.
{'Country': {0: 'Austria', 1: 'Belgium', 2: 'Denmark', 3: 'Finland', 4: 'France', 5: 'Germany', 6: 'Iceland', 7: 'Ireland', 8: 'Italy', 9: 'Luxemburg', 10: 'Netherland', 11: 'Norway', 12: 'Portugal', 13: 'Spain', 14: 'Sweden', 15: 'Switzerland', 16: 'United Kingdom'}, 'y1989': {0: 7602431, 1: 9927600, 2: 5129800, 3: 4954359, 4: 56269800, 5: 61715000, 6: 253500, 7: 3526600, 8: 57504700, 9: 374900, 10: 14805240, 11: 4226901, 12: 10304700, 13: 38851900, 14: 8458890, 15: 6619973, 16: 57236200}, 'y1990': {0: 7660345.0, 1: 9947800.0, 2: 5135400.0, 3: 4974383.0, 4: 0.0, 5: 62678000.0, 6: 255708.0, 7: 3505500.0, 8: 57576400.0, 9: 379300.0, 10: 14892574.0, 11: 4241473.0, 12: 0.0, 13: 38924500.0, 14: 8527040.0, 15: 6673850.0, 16: 57410600.0}, 'y1991': {0: 7790957, 1: 9987000, 2: 5146500, 3: 4998478, 4: 56893000, 5: 79753000, 6: 259577, 7: 3519000, 8: 57746200, 9: 384400, 10: 15010445, 11: 4261930, 12: 9858500, 13: 38993800, 14: 8590630, 15: 6750693, 16: 57649200}, 'y1992': {0: 7860800, 1: 10068319, 2: 5162100, 3: 5029300, 4: 57217500, 5: 80238000, 6: 262193, 7: 3542000, 8: 57788200, 9: 389800, 10: 15129200, 11: 4273634, 12: 9846000, 13: 39055900, 14: 8644100, 15: 6831900, 16: 58888800}, 'y1993': {0: 7909575, 1: 10100631, 2: 5180614, 3: 5054982, 4: 57529577, 5: 81338000, 6: 264922, 7: 3559985, 8: 57114161, 9: 395200, 10: 15354000, 11: 4324577, 12: 9987500, 13: 39790955, 14: 8700000, 15: 6871500, 16: 58191230}, 'y1994': {0: 7943652, 1: 10130574, 2: 5191000, 3: 5098754, 4: 57847000, 5: 81353000, 6: 266783, 7: 3570700, 8: 57201800, 9: 400000, 10: 15341553, 11: 4348410, 12: 9776000, 13: 39177400, 14: 8749000, 15: 7021200, 16: 58380000}, 'y1995': {0: 8054800, 1: 10143047, 2: 5251027, 3: 5116800, 4: 58265400, 5: 81845000, 6: 267806, 7: 3591200, 8: 57268578, 9: 412800, 10: 15492800, 11: 4370000, 12: 9920800, 13: 39241900, 14: 8837000, 15: 7060400, 16: 58684000}}
My code
df[(df.Country)& (df.y1989>1000000)]
Error:
TypeError: unsupported operand type(s) for &: 'str' and 'bool'
I am not sure what could be the reason, being a newbie to python if you could provide explanation for the error that will be greatly appreciated.
Thanks in advance,

'Country' doesn't form part of your filtering criteria, so don't use it to form your Boolean indexer. Instead, use the loc accessor to give a Boolean condition and specify necessary columns separately:
res = df.loc[df['y1989'] > 1000000, ['Country','y1989']]
Under no circumstances use chained assignment, e.g. via df[df['y1989']>1000000][['Country','y1989']], as this is ambiguous and explicitly discouraged in the docs.