Selection of 4 Balls with At Least 2 Red Balls from a Set of 4 Red and 5 Pink Balls
This article delves into a classic combinatorics problem: determining the number of ways to select 4 balls such that at least 2 of them are red, from a set of 4 red balls and 5 pink balls. We'll explore all possible scenarios and calculate the total number of selections.
Introduction
Combinatorics is a branch of mathematics that deals with the selection, arrangement, and operation of elements in sets. In this specific problem, we are tasked with selecting 4 balls out of a total of 9 (4 red and 5 pink) with the constraint that at least 2 of the selected balls must be red. Let's break down the problem using combinatorial methods and ensure the calculations are accurate.
Breakdown of Selection Scenarios
To solve this problem, we need to consider different cases based on the number of red balls in the selection. We'll explore each case in detail:
Scenario 1: 4 Red Balls and 0 Pink Balls
The first scenario involves selecting 4 red balls out of the 4 available. This selection can be done in only one way, since all red balls must be chosen:
.Selection 1 way
Scenario 2: 3 Red Balls and 1 Pink Ball
In this case, we choose 3 red balls from the 4 available (4 choose 3) and 1 pink ball from the 5 available (5 choose 1). The total number of ways to achieve this is calculated as follows:
Number of ways 4 choose 3 × 5 choose 1 4 × 5 20 ways
Scenario 3: 2 Red Balls and 2 Pink Balls
Here, we need to select 2 red balls (4 choose 2) and 2 pink balls (5 choose 2). The calculation is as follows:
Number of ways 4 choose 2 × 5 choose 2 6 × 10 60 ways
Total Number of Selections
By summing up the number of ways for each scenario, we can determine the total number of selections where at least 2 of the 4 balls are red:
Total 1 (from Scenario 1) 20 (from Scenario 2) 60 (from Scenario 3) 81 ways
Conclusion
Through a combination of combinatorial methods and careful calculation, we can confidently state that there are 81 different ways to select 4 balls with at least 2 red balls from a set of 4 red and 5 pink balls. This problem demonstrates the power of combinatorics in solving selection problems.
Additional Insights
Understanding combinatorial principles, such as permutations and combinations, is crucial in various fields, including statistics, computer science, and data analysis. By practicing similar problems, one can develop a deeper appreciation for the beauty and utility of mathematical concepts in real-world scenarios.