3.2 The product rule

The product rule applies when options are sequential, and they are joined by a conjunction.

For example, let us consider the number of possibilities of picking two cards from a deck of 10 distinct cards. There are 10 possibilities for the first pick, and there are 9 possibilities for the second pick. Because the two picks are independent, the total number of possibilities is $10\cdot 9 = 90$ . In other words, if we are to pick two cards from a deck of 10, there are 90 possible ways. Note that in this case, the order of picking is important. The card sequence (1, 2) is distinct from that of (2, 1).

The product rule is linked with conjunction (``and''). In our example, there are 10 cards to choose from for the first card, and there are 9 cards to choose from for the second card. You can also think that the product rule deals with sequences.

We can also put everything in more formal terms. Let $X_i$ represent the set of possibilities in option $i$ , and there are $n$ options to be applied sequentially. The set of all possible sequences is the Cartesian product $\prod_{i=1}^{n} X_i$ . The total number of possibilities is $\vert\prod_{i=1}^{n} X_i\vert = \prod_{i=1}^{n} \vert X_i\vert$ . (We used the same symbol $\prod$ for arithmatic multiplication and Cartesian product.)