Understanding the Number of Ways to Select at Least One Fruit from a Variety

Introduction

When dealing with combinations and selections in mathematics and combinatorics, it's essential to understand various scenarios and rules. One common problem is to find the number of ways in which at least one fruit can be selected from a given variety. In this article, we will explore this concept with a specific example and provide a step-by-step explanation.

Problem statement and Initial Analysis

What are the number of ways in which at least one fruit can be selected from 5 oranges, 4 apples, and 3 bananas?"" we."" We will use the principle of counting combinations to solve this problem.

Step 1: Calculate Total Combinations Including the Option of Selecting None

Oranges: You can select from 0 to 5 oranges. This gives you 6 options (0, 1, 2, 3, 4, 5). Apples: You can select from 0 to 4 apples. This gives you 5 options (0, 1, 2, 3, 4). Bananas: You can select from 0 to 3 bananas. This gives you 4 options (0, 1, 2, 3).

The total combinations of selecting fruits including selecting none is given by:

6 oranges times; 5 apples times; 4 bananas 120.

Number of ways to select no fruit is 1 (selecting 0 oranges, 0 apples, and 0 bananas).

Step 2: Subtract the Case Where No Fruit is Selected

From the total combinations of 120, we subtract the case where no fruit is selected:

120 - 1 119.

Therefore, the number of ways to select at least one fruit from 5 oranges, 4 apples, and 3 bananas is 119.

Condition: Fruits of each species are identical and non-identical

When the fruits are identical, the number of ways to select at least one fruit from each type is:

5 times; 4 times; 3 60.

If the fruits are not identical, we use the formula for combinations:

2^5 - 1 31 for oranges, 2^4 - 1 15 for apples, and 2^3 - 1 7 for bananas.

The total number of combinations is:

31 times; 15 times; 7 3255.

Alternative Solution

Another approach is to first select one fruit from each type, leaving 2 more fruits to be selected. We then use the concept of combinations to determine the number of ways to choose the remaining 2 fruits from the remaining 12 (3 oranges - 1, 2 bananas - 1, 1 apple - 1).

The remaining 12 fruits can be chosen in:

C(12, 2) (12 times; 11) / (2 times; 1) 66.

Since we have already chosen 3 fruits (one from each type), the total number of ways is:

66 1 67.

Adding the initial 3 selected fruits, the total is 67.

Conclusion

The problem of selecting at least one fruit from a given variety can be solved using the principle of counting combinations. By carefully analyzing the different cases, we can derive the correct number of ways. Whether the fruits are identical or not has a significant impact on the number of combinations.

Understanding these principles helps in solving similar problems in combinatorics and other related fields.