Question

shoppers in a large shopping mall are categorized as male or female, over 30 or 30 and under, and cash or credit card shoppers. in how many ways can the shoppers be categorized?
the shoppers can be categorized in \\(\square\\) different ways.

Explanation:

Step1: Use the multiplication principle

The number of ways to categorize by gender is \(2\) (male or female). The number of ways to categorize by age is \(2\) (over 30 or under 30). The number of ways to categorize by payment method is \(2\) (cash or credit card).

Step2: Calculate the total number of ways

By the multiplication principle of counting, if there are \(m_1\) ways to do one thing, \(m_2\) ways to do a second thing, and \(m_3\) ways to do a third thing, the total number of ways to do all three things together is \(m_1\times m_2\times m_3\). Here, \(m_1 = 2\), \(m_2=2\), \(m_3 = 2\). So the total number of ways \(N=2\times2\times2\).
$$N = 2\times2\times2=8$$

Answer:

\(8\)