Combinatorial Analysis in Probability: Calculating the Number of Ways to Draw Specific Balls from a Box

In probability theory and combinatorial mathematics, it is often necessary to calculate the number of ways to achieve specific outcomes, particularly when dealing with drawing balls from a box. This article will explore the method to find the number of ways to draw 2 blue balls and 1 red ball from a box containing 6 red balls, 5 green balls, and 3 blue balls, in a particular context.

Understanding the Problem

Consider a box containing 6 red balls, 5 green balls, and 3 blue balls. Our task is to determine the number of ways to draw 3 balls from this box, with the condition that the result must consist of exactly 2 blue balls and 1 red ball. This scenario can be analyzed using combinatorial methods, specifically combination and permutation principles.

Combinatorial Principles

The key principles in this problem are the combination formula and the permutation principle.

Combination Formula

The combination formula is used to count the number of ways to choose a subset of items from a larger set without regard to the order of selection. The formula is given by:

C(n, k) n! / [k!(n - k)!]

where n is the total number of items, and k is the number of items to choose.

Applying the Combinatorial Formula

Let's apply this to our specific scenario.

Number of Ways to Choose 1 Red Ball

There are 6 red balls in the box, so the number of ways to choose 1 red ball is:

C(6, 1) 6! / [1!(6 - 1)!] 6

Number of Ways to Choose 2 Blue Balls

There are 3 blue balls in the box, so the number of ways to choose 2 blue balls is:

C(3, 2) 3! / [2!(3 - 2)!] 3

Total Number of Arrangements

If we consider that the order of drawing the balls is important, each set of chosen balls (1 red and 2 blue) can be arranged in 3! (3 factorial) different ways. The value of 3! is 6, because 3! 3×2×1 6.

Multiplying the Combinations

To find the total number of ways to achieve the desired outcome, we multiply the number of ways to choose the red ball by the number of ways to choose the blue balls and by the number of permutations:

Total arrangements C(6, 1) × C(3, 2) × 3!

6 × 3 × 6 108

Conclusion

Thus, there are 108 distinct ways to draw 2 blue balls and 1 red ball from a box containing 6 red balls, 5 green balls, and 3 blue balls, considering that the order of drawing is significant.

Additional Considerations

It is essential to recognize that this problem assumes distinct balls. If the balls are indistinguishable, the problem would require a different method of calculation, such as generating functions or other advanced combinatorial techniques.