Bernoulli IV

Part Three.
The use of the Preceding Doctrine in Various Ways of Casting Lots and Games of Chance.
“Problem V”
“A contends with B that from 40 playing cards, that is, 10 of each suit, he will choose 4 cards such that there is one of each suit. What is the ratio of their lots?”
“To this end it is asked in how many ways 4 playing cards may be chosen from 40, that is, it is asked what is the number of quaternions in 40 things.”
The solution presented makes use of the Doctrine of Combinations from Part two.
For the first part we see from Part Two Chapter IV that there are = 91,390 possible combinations of ways to draw 4 cards from 40. However, this only takes into account being able to draw combinations that may have the same suit represented twice. In order to find out how many combinations there are when all suits are represented once, Bernoulli proposes an image of 4 ten-sided dice, with the 4 representing the number of suits and the 10 sides representing the number of cards in each suit. Now we see that the number of combinations that can be drawn when all suit are represented once will be the same as the number of possible throws with these dice. Namely: 10∙10∙10∙10 throws which equals 10,000 throws. Now to find the Ratio of their lots we see that if A has 10,000 combinations which will produce his desired lot, his opponent, B has 91,390-10,000 = 81,390 possible combinations which will NOT contain A’s desired outcome. Therefore, the ratio of their lots is 10,000 to 81,390 or 1,000 to 8,139.