Selecting Balls from a Box: A Combinatorial Approach and Google SEO Optimization
In this article, we explore the problem of selecting 5 balls from a box containing 4 red, 3 blue, and 5 yellow balls, with the specific condition that exactly 3 of the selected balls are red. We'll use combinatorial methods to solve this problem and discuss how such content can be optimized for Google SEO.Problem Statement and Solution
We have a box containing 4 red, 3 blue, and 5 yellow balls. The task is to determine the number of ways to select 5 balls such that exactly 3 of them are red. We break this problem into two parts: selecting the red balls and selecting the remaining non-red balls.Selecting the Red Balls
Since we need exactly 3 red balls out of the 4 available, we use the combination formula (binom{n}{r}) to find the number of ways to choose 3 red balls from 4. This is calculated as follows: [binom{4}{3} frac{4!}{3!(4-3)!} frac{4}{1} 4]Selecting the Remaining Balls
After choosing 3 red balls, we need to select 2 more balls from the remaining 8 balls (1 red, 3 blue, and 5 yellow). The number of ways to choose 2 balls from these 8 is: [binom{8}{2} frac{8!}{2!(8-2)!} frac{8 times 7}{2 times 1} 28]Combining the Selections
To find the total number of ways to select the 5 balls with the specified condition, we multiply the number of ways to choose the red balls by the number of ways to choose the remaining balls: [text{Total ways} binom{4}{3} times binom{8}{2} 4 times 28 112] Therefore, the total number of ways to select 5 balls such that there are exactly 3 red balls is 112.Alternative Approach to Combinatorial Selection
Another way to approach the problem is to consider the different combinations of the remaining balls after choosing 3 red balls. The remaining balls can be either 2 blue, 2 yellow, or 1 blue and 1 yellow. There are 3 such combinations, but this approach does not provide the exact count, just a range of possibilities. The combinatorial method used in the initial problem statement gives a precise count of 112 ways.Google SEO Optimization
For this article to be optimized for Google SEO, we need to focus on several key points.Keyword Usage
The primary keywords for this content are 'combinatorial selection' and 'red balls selection'. Using these keywords naturally in the content can help improve the relevance and ranking in search engine results. Additionally, incorporating variations like 'ball selection' and 'combinatorics' can also be beneficial.Meta Tags and Descriptions
Ensure that the tags include the main keywords: []Header Tags
Use header tags effectively to structure the content and make it easier for Google to understand the hierarchy of information. For instance, the section headers could follow this pattern: [Selecting Balls from a Box: A Combinatorial Approach and Google SEO Optimization
Problem Statement and Solution
Selecting the Red Balls
Selecting the Remaining Balls
Combining the Selections
Alternative Approach to Combinatorial Selection
Google SEO Optimization
]Content Surrounding Keywords
Make sure keywords are integrated into the main content, such as in the problem statement, solution, and alternative approaches. For example: [For the red balls selection, we use the combination formula to find the number of ways to choose exactly 3 red balls from the available 4.
]