Question

If the first event can occur in m ways and the second event can occur in n ways, what is the total number of ways that the two events can occur?

1) m + n
2) m - n
3) m * n
4) m / n

Answer 1

The total number of ways two independent events can occur if the first event can happen in m ways and the second in n ways is m * n. This is the multiplication principle of counting in combinatorics. Option 3, m * n, is the correct answer.

Step-by-step explanation:

The question is regarding the fundamental principle of counting in probability and combinatorics. If the first event can occur in m ways and the second event can occur in n ways, and the two events are independent, then the total number of ways both events can occur is found by multiplying the number of ways the first event can occur by the number of ways the second event can occur. Hence, the answer to this question is m * n.

Let's consider a simple example to illustrate this concept. If you have 3 different shirts (m = 3) and 4 different pants (n = 4), then the number of different outfits (combinations of shirts and pants) you can create is 3 shirts * 4 pants = 12 outfits. This multiplication principle applies to any two independent events in combinatorics.

The multiplication rule we've described is not to be confused with other operations such as addition, subtraction, or division, which are applied under different circumstances in probability and combinatorics. For this reason, options 1 (m + n), 2 (m - n), and 4 (m / n) are not correct for this specific question.