How Many Ways Are There to Draw 2 Balls Out of 3 White Balls? Unraveling the Combination Formula Its Significance

Have you ever wondered about the number of ways you can draw 2 balls from a set of 3 white balls? It's a question that delves into the realms of combinatorics, which is the branch of mathematics that deals with counting, arrangement, and combination of objects. Let’s explore the solution using the combination formula and understand its implications.

Understanding Combinations

Combinations are a fundamental concept in mathematics, particularly in statistics and probability. The combination formula, also known as the binomial coefficient, is used to calculate the number of ways in which a subset of items can be selected from a larger set without regard to the order of selection. The formula is given by:

(binom{n}{r} frac{n!}{r! (n-r)!})

This formula can be broken down into simpler terms, where:

n is the total number of items available, r is the number of items to be selected.

In the context of our example, we have 3 white balls and we want to select 2 of them. Therefore, n 3 and r 2. Plugging these values into the formula, we get:

(binom{3}{2} frac{3!}{2! (3-2)!} frac{3!}{2! cdot 1!} frac{3 times 2 times 1}{2 times 1 times 1} frac{6}{2} 3)

Thus, there are 3 ways to select 2 balls from 3 white balls. The three combinations are:

Ball 1 and Ball 2 Ball 1 and Ball 3 Ball 2 and Ball 3

Why the Answer is 3, Not 1

The answer is indeed 3, not 1, because each combination represents a unique selection of balls, even though the balls themselves are identical. This is a crucial aspect of combinations - the order of selection does not matter. Each pair is considered distinct regardless of which ball is selected first or second.

Consider a similar scenario where you need to select 2 green balls and 1 red ball from a bag containing 5 green and 5 red balls. The answer would be (binom{5}{2} times binom{5}{1}). Similarly, in the case of white balls, each way of selecting the balls represents a unique combination, hence the answer is 3.

Visualizing Combinations with Identical Objects

Imagine you have three toys: a car, a plane, and a tank. You want to pick two toys from the group. The number of ways to do this can be calculated as (binom{3}{2}). You can easily see that the possible combinations are:

Car and Plane Car and Tank Plane and Tank

This demonstrates that even with identical white balls, if we consider each ball with a unique marker (e.g., numbered 1, 2, and 3), the possible pairs become:

1 and 2 1 and 3 2 and 3

Therefore, the number of combinations is 3, consistent with the combination formula.

Additional Insights into Combinations

Let’s delve deeper into the nuances of combinations through two specific cases:

Understanding nC0

nC0 is the number of ways to select 0 items from a set of n items. By definition, there is only one way to do this - by selecting nothing. Mathematically, this is represented as:

(binom{n}{0} 1)

This makes sense because no matter how you choose to select 0 items, the result is always the same - no selection. Hence, only one way is possible.

Understanding nCn

nCn is the number of ways to select all n items from a set of n items. Again, by definition, there is only one way to do this - by selecting all the items. Mathematically, this is represented as:

(binom{n}{n} 1)

This makes sense because no matter how you choose to select all items, the result is always the same - all items. Hence, only one way is possible.

Conclusion

Through this exploration, we have seen that the number of ways to draw 2 balls out of 3 white balls is indeed 3, not 1. This is because combinations are concerned with the selection of items, where the order does not matter. Each unique combination of selected items is a distinct outcome.

Understanding combinations and permutations is crucial in various fields, including statistics, probability, and computer science. Whether you are dealing with identical objects or distinct objects, the principles remain the same, making combinations an essential tool in mathematical problem-solving.