Combinations and Permutations: Selecting Two Balls from Three

Imagine a scenario where you have three balls, each distinctly labeled. The challenge is to determine the number of ways to select any two of those balls. This scenario can be solved using the fundamental principles of combinations and permutations. Let's explore this concept in detail.

Understanding the Problem

In this scenario, we have three balls: Ball 1, Ball 2, and Ball 3. The goal is to find out how many different ways we can choose two balls from these three.

Solution

The solution involves the concept of combinations. We can apply the formula for combinations to determine the number of ways to select two balls from three.

Formula for Combinations

The formula for combinations is given by:

[ ^nC_r frac{n!}{r!(n-r)!} ]

where ( n ) is the total number of objects, ( r ) is the number of objects to be chosen, and ( ! ) denotes the factorial of a number.

Applying the Formula

In our scenario, we have ( n 3 ) and ( r 2 ). Plugging these values into the formula, we get:

[ ^3C_2 frac{3!}{2!(3-2)!} ]

Calculating the factorials, we have:

[ 3! 3 times 2 times 1 6 ]

[ 2! 2 times 1 2 ]

[ (3-2)! 1! 1 ]

Substituting these values back into the formula:

[ ^3C_2 frac{6}{2 times 1} 3 ]

Hence, the number of ways to select two balls from three is 3.

Interpreting the Solution

The result of 3 indicates that there are three distinct combinations of two balls that can be selected from three:

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

This can also be understood by considering the opposite: if one ball is left out, there are three possibilities—leaving out Ball 1, Ball 2, or Ball 3. Each of these scenarios corresponds to one of the combinations mentioned above.

Conclusion

The problem of selecting two balls from three is a classic example of combinations in mathematics. By applying the combination formula, we can determine that the number of ways to choose two balls from three is 3. This concept is useful in various fields, including statistics, probability, and computer science.

Further Reading

For a deeper understanding of combinations and permutations, you might want to explore:

A deeper dive into the binomial theorem Applications in probability theory Permutations and their uses in cryptography

Understanding these concepts can provide a strong foundation in mathematical problem-solving.