Number of ways to select at least one fruit from the given options.

There are 5 oranges, 4 apples, and 3 bananas. We need to find the number of ways in which we can select at least one fruit from these options.

Step 1: Find the number of ways to select fruits from each category.

To find the number of ways to select fruits from each category, we can use the concept of combinations. The number of ways to select 'r' items from a set of 'n' items is given by the formula:
nCr = n! / (r!(n-r)!)

Let's calculate the number of ways to select fruits from each category:

- Oranges: There are 5 oranges, so we can select 0, 1, 2, 3, 4, or 5 oranges. Therefore, the number of ways to select oranges is:
5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5

- Apples: There are 4 apples, so we can select 0, 1, 2, 3, or 4 apples. Therefore, the number of ways to select apples is:
4C0 + 4C1 + 4C2 + 4C3 + 4C4

- Bananas: There are 3 bananas, so we can select 0, 1, 2, or 3 bananas. Therefore, the number of ways to select bananas is:
3C0 + 3C1 + 3C2 + 3C3

Step 2: Find the total number of ways to select at least one fruit.

To find the total number of ways to select at least one fruit, we need to consider all possible combinations of selecting fruits from each category. We can do this by multiplying the number of ways to select fruits from each category.

Let's calculate the total number of ways to select at least one fruit:

Total ways = (5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5) * (4C0 + 4C1 + 4C2 + 4C3 + 4C4) * (3C0 + 3C1 + 3C2 + 3C3)

Simplifying the expression, we get:

Total ways = (1 + 5 + 10 + 10 + 5 + 1) * (1 + 4 + 6 + 4 + 1) * (1 + 3 + 3 + 1)

Total ways = 32 * 16 * 8

Total ways = 4096

Therefore, there are 4096 ways in which at least one fruit can be selected from the given options.

Conclusion:

In this problem, we used the concept of combinations to calculate the number of ways to select fruits from each category. Then, we multiplied these individual values to find the total number of ways to select at least one fruit from the given options. The final answer is 4096.