Rule of Product Applied to Decks of Cards

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If there are ways to do and ways to do no matter what the result was for , then the number of ways to do both and is .


In this Demonstration, is the result of drawing a card of a given suit and is the result of drawing a card of a given value (1, 2, 3, …, J, Q, K, A). The suit of the card drawn has no influence on its value, so the product rule applies.

Here is another example (not shown in the graphic). How many ways are there of first drawing a red card and then drawing a spade (♠)? The two sets are reds and spades; they are distinct because a spade is black. There are 26 ways to draw a red card and 13 ways to draw a spade, so the number of possible ways to draw a red card and then a spade is .


Contributed by: S. M. Blinder (July 2018)
Open content licensed under CC BY-NC-SA



The product rule generalizes when there are more than two kinds of results, as long as they are pairwise independent. So .


[2] Wikipedia. "Rule of Product." (Jun 4, 2018)

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