To determine the number of ways to select fruits from a crate containing 6 apples, 8 oranges, and 12 pineapples, we address two scenarios. Firstly, if at least one pineapple must be chosen, we utilize complementary counting by calculating the total number of ways to select 5 fruits and subtracting the number of ways to choose 5 fruits without selecting any pineapples. This ensures that at least one pineapple is included in the selection. Secondly, if at least two different fruits must be chosen, we employ the principle of inclusion-exclusion. We calculate the total number of ways to select 5 fruits and subtract the number of ways to choose 5 fruits with only one type of fruit (apples, oranges, or pineapples) selected. This ensures that the selection comprises at least two different types of fruits. Through these systematic approaches, we ascertain the number of feasible selections meeting the specified criteria, thereby facilitating effective decision-making regarding fruit consumption from the crate.
