Problem 80 - Algebra and Trigonometry

Question

Complementary Combinations Without performing any calculations, explain in words why the number of ways of choosing two objects from ten objects is the same as the number of ways of choosing eight objects from ten objects. In general, explain why $$ C(n, r)=C(n, n-r) $$

Step-by-Step Solution

Answer

Choosing 2 out of 10 is the same as not choosing 8 out of 10. Thus, \(C(10, 2) = C(10, 8)\).

Step 1: Understand the Problem

We need to explain why the number of ways to choose 2 objects from 10 is the same as the number of ways to choose 8 objects from 10. This question explores the concept of complementary combinations.

Step 2: Introduction to Combinations

Combinations formula is given by the binomial coefficient: \(C(n, r) = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of objects and \(r\) is the number of selections.

Step 3: Definition of Complementary Selection

If you choose 2 objects from a set of 10, you are simultaneously not choosing the remaining 8 objects. Therefore, choosing 2 objects is equivalent to not choosing 8 objects.

Step 4: General Rule for Complementary Combinations

In general, the rule \(C(n, r) = C(n, n-r)\) signifies that the number of ways to choose \(r\) objects from \(n\) is the same as choosing \(n-r\) objects and leaving \(r\) objects unchosen.

Step 5: Apply Formula to Example

Using the formula, \(C(10, 2) = \frac{10!}{2!(10-2)!} = C(10, 8) = \frac{10!}{8!(10-8)!}\), we can see both calculations evaluate the same number because the mathematical structure is symmetric around \(n/2\) in the context of combinations.

Key Concepts

Complementary Combinations

Complementary combinations help us understand that choosing is not just about selecting specific items, but also about what we leave behind. When asked to choose a subset from a larger set, if you choose some items, the rest remain unselected. For example, choosing 2 objects from 10 leaves 8 objects unchosen. In this way, both choices are equally significant. This realization allows us to see that the number of ways to choose a subset of objects corresponds directly to the number of ways of not choosing the remaining objects from the same pool. Complementary combinations reveal that the act of choosing is also about what is omitted, providing a new perspective on selection problems.

Binomial Coefficient

The binomial coefficient offers a way to calculate combinations effectively. It is notated as \(C(n, r)\) and calculates how many ways we can choose \(r\) objects from \(n\) total objects. The formula for the binomial coefficient is: \ - \( C(n, r) = \frac{n!}{r!(n-r)!} \) \ This formula considers all potential arrangements of \(r\) objects and then eliminates any redundant sequences created by rearrangements. The factorial \(n!\) represents all possible arrangements, while \(r!\) and \((n-r)!\) adjust for overcounted permutations. This consistent mathematical tool is essential for combinatorial calculations.

Combinations Formula

The combinations formula is a fundamental principle in combinatorics, used to calculate the number of ways to select items from a set. It relies on the idea that the order of selection doesn’t matter. The formula involved is: \ - \( C(n, r) = \frac{n!}{r!(n-r)!} \) \ This formula helps simplify problems where we need to make selections from larger groups, offering insights into a wide range of applications. Each variable in the formula has a particular purpose: \

• \(n\) stands for the total number of items in the set.
• \(r\) represents the number of items selected.
• The difference \((n-r)\) calculates how many remain unselected.

This approach ensures precise counting of all possible combinations, central to solving various practical problems.

Combinatorial Symmetry

Combinatorial symmetry is a fascinating concept which indicates that combination calculations share properties that make them mirror-like in specific scenarios. Specifically, for any set size \(n\), choosing \(r\) items leaves \(n-r\) items unchosen, giving rise to the formula \(C(n, r) = C(n, n-r)\). This symmetry highlights that the process of forming combinations can be seen in two equivalent manners. Consider how symmetrical this is when viewing choices from two perspectives:

• Counting chosen items.
• Counting unchosen items.

The equating of \(C(10, 2)\) and \(C(10, 8)\) demonstrates this symmetry elegantly, emphasizing the natural balance present in combinatorial problems. This concept is not only mathematically elegant but also immensely useful, offering double the insight into any problem involving selections within sets.