Final answer:
By applying the principle of counting, we can multiply the number of sleeve options (2) by the number of color options (4) to get 8 different styles of shirts with school logos.
Step-by-step explanation:
To determine the number of different styles of shirts that the pep club can sell, we can use the fundamental principle of counting. This principle states that if there are two independent choices to be made and the first choice can be made in m ways and the second choice can be made in n ways, then the total number of different ways to make both choices is m x n. In this scenario, there are two types of sleeves (long sleeves and short sleeves), and four color choices (orange, green, purple, and blue).
The number of different styles can be calculated by multiplying the number of sleeve options by the number of color options:
Number of sleeve options (2) x Number of color options (4) = 8 different styles.
Therefore, the pep club can sell a total of 8 different styles of shirts.