The question is asking how many ways we can select at least one fruit of each type from three distinct oranges, four distinct apples, and five distinct mangoes.

Interpretation:

1. There are 3 distinct oranges.
2. There are 4 distinct apples.
3. There are 5 distinct mangoes.

You need to select at least one of each type. For each fruit type, the choices are as follows:

• Oranges: We can choose either 1, 2, or all 3 oranges. This gives $2^3 - 1 = 7$ possibilities (since we exclude the case where none is selected).
• Apples: We can choose either 1, 2, 3, or all 4 apples. This gives $2^4 - 1 = 15$ possibilities.
• Mangoes: We can choose either 1, 2, 3, 4, or all 5 mangoes. This gives $2^5 - 1 = 31$ possibilities.

Total number of ways:

Since these choices are independent for each fruit type, we multiply the possibilities together:

$7 \times 15 \times 31 = 3255$

Thus, the total number of ways to select at least one fruit of each type is 3,255.

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:

1. What happens if the fruits are not distinct?
2. How would the answer change if you could select zero fruits from some types?
3. How do combinations and permutations differ in this context?
4. What would be the total number of ways if there were only 2 types of fruits?
5. How would you calculate if the number of fruit types increased to 4 or more?

Tip: Problems like this are solved by using the principle of inclusion-exclusion when avoiding the "no selection" case!