Problem 44 - Algebra 2 and Trigonometry

Question

Show that \(_{n} C_{r}=_{n} C_{n-r}\)

Step-by-Step Solution

Answer

\(_{n} C_{r} = _{n} C_{n-r}\) because \(\frac{n!}{r! (n-r)!} = \frac{n!}{(n-r)! r!}\).

Step 1: Understanding the Problem

We need to show that the number of ways to choose \( r \) items from \( n \) items is the same as choosing \( n-r \) items from \( n \) items. This is a symmetry property of combinations.

Step 2: Using the Formula for Combinations

Recall the formula for combinations: \( _{n} C_{r} = \frac{n!}{r! (n-r)!} \). This represents the number of ways to choose \( r \) items from \( n \) items.

Step 3: Substitute \( n-r \) for \( r \) in the Formula

Substitute \( n-r \) for \( r \) in the combination formula: \( _{n} C_{n-r} = \frac{n!}{(n-r)! (n-(n-r))!} = \frac{n!}{(n-r)! r!} \).

Step 4: Compare the Two Expressions

Notice that both expressions, \( _{n} C_{r} = \frac{n!}{r! (n-r)!} \) and \( _{n} C_{n-r} = \frac{n!}{(n-r)! r!} \), are identical because multiplication is commutative.

Key Concepts

Symmetry Property of Combinations

The symmetry property of combinations is a fascinating aspect of combinatorics. This property highlights that choosing a subset of items from a larger set can be viewed in two mirrored ways. Let's break it down: when you select \( r \) items from a total of \( n \) items, it is essentially the same as leaving out \( r \) items and selecting the remaining \( n-r \).

Imagine you have a box of colorful marbles and you decide to take three out of the ten. According to the symmetry property, this choice is equivalent to deciding which seven marbles will stay in the box. Each decision results in the complementary set staying behind or being chosen.

• The focus is on the remaining or left items.
• It reflects that both sets, the chosen and the unchosen, are equally significant combinations.

Understanding the symmetry property simplifies solving complex problems by providing different perspectives.

Formula for Combinations

Combinations are used in mathematics to determine the number of ways to choose a set of elements without considering the order. The formula for combinations is given by:

• \( _{n} C_{r} = \frac{n!}{r! (n-r)!} \)
• It represents the total number of ways to choose \( r \) items from \( n \).

This formula is derived from the basic principles of permutation and division to remove the factor of order. Essentially, when you're choosing \( r \) items from \( n \), you want to count only the combinations, not permutations, hence dividing by permutations of the chosen and unchosen items.

For example, if you wanted to choose 3 toppings out of 5 available for a pizza, you would use this formula to discover all possible unique topping combinations without repetition. This approach helps in various fields like probability, statistics, and operational research.

Factorial Notation

A fundamental part of the combinations formula is factorial notation. Factorial notation, denoted as \( n! \), means multiplying all positive integers up to \( n \). For instance, \( 5! \) equals \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

• It represents the product of all positive integers less than or equal to a specific number.
• Factorials provide a mathematical way to express large numbers succinctly.

Factorials are pivotal in combinatorics, as they simplify expressions within the combinations formula.

Whenever you see a factorial in combinations or permutations, it depicts arranging or choosing options.For instance, in a deck of cards, the total number of arrangements can be calculated as \( 52! \), representing each unique arrangement of cards in a deck. Using factorials in the combinations formula allows us to efficiently compute large selection problems without manually counting each possibility.