Probability and Combinations in Card Selection: A Detailed Analysis

In this article, we will delve into the mathematical concepts of combinations and probability. Specifically, we will calculate the total number of ways to pick 3 cards from a box containing 3 blue, 2 green, 1 red, and 4 white cards. Additionally, we will explore the number of ways to pick one green card and two white cards.

Introduction to the Card Box

The box contains a total of 10 cards: 3 blue, 2 green, 1 red, and 4 white. We will consider the scenarios of picking 3 cards at random and determine the number of possible combinations.

Total Number of Ways to Pick 3 Cards from the Box

When picking 3 cards from a box containing 10 cards one at a time without replacement, we can calculate the total number of ways using the combination formula C(n, k) n! / (k!(n-k)!), where n is the total number of cards and k is the number of cards being selected.

Total number of combinations:

Using the combination formula, the total number of ways to pick 3 cards from 10 is:

$$C(10, 3) frac{10!}{3!(10-3)!} frac{10 times 9 times 8}{3 times 2 times 1} 120$$

Therefore, there are 120 ways to pick 3 cards from the box.

Number of Ways to Pick 1 Green and 2 White Cards

Next, we will calculate the number of ways to pick exactly 1 green card and 2 white cards from the box.

Ways to pick 1 green card:

There are 2 green cards, so the number of ways to pick 1 green card is:

$$C(2, 1) frac{2!}{1!(2-1)!} 2$$

Ways to pick 2 white cards:

There are 4 white cards, so the number of ways to pick 2 white cards is:

$$C(4, 2) frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6$$

Total number of ways to pick 1 green and 2 white cards:

By multiplying the number of ways to pick 1 green card by the number of ways to pick 2 white cards, we get:

$$2 times 6 12$$

Therefore, there are 12 ways to pick 1 green card and 2 white cards from the box.

Conclusion

In this article, we have explored the principles of combination and applied them to a practical scenario involving picking cards from a box. We calculated the total number of ways to pick 3 cards from a box containing 10 cards and determined the number of ways to pick 1 green card and 2 white cards. Understanding these concepts can be valuable in various real-world applications, such as probability and statistics.

Key Takeaways:

The total number of ways to pick 3 cards from a box containing 10 cards is 120. The number of ways to pick 1 green card and 2 white cards from the box is 12.

For further reading, you can explore more detailed resources on probability and combinatorics. This knowledge can be invaluable in fields such as data analysis, cryptography, and game theory.