Problem 101 - Precalculus with Limits

Question

In Exercises 101-104, prove the property for all integers \( r \) and \( n \) where \( 0 \le r \le n \). \( _nC_r = _nC_{n - r} \)

Step-by-Step Solution

Answer

The property \( _nC_r = _nC_{n - r} \) is proved through the formula for binomial coefficients and algebraic simplification, showing that choosing \( r \) items out of \( n \) is same as choosing \( n - r \) items out of \( n \) for all integers \( r \) and \( n \) where \( 0 \le r \le n \).

Step 1: Formula of Binomial Coefficients

Recall the formula of binomial coefficients, expressed as \( _nC_r = \frac{n!}{r!(n - r)!} \) where \( n! \) means 'n factorial', that is the product of all positive integers up to \( n \), and similarly for \( r \) and \( n - r \).

Step 2: Write both sides of the equality

Write \( _nC_r = _nC_{n - r} \) as \( \frac{n!}{r!(n - r)!} = \frac{n!}{(n - r)!r!} \)

Step 3: Simplify the equation

Since both sides of the equation have the same numerator \( n! \), the equation simplifies to \( \frac{1}{r!(n - r)!} = \frac{1}{(n - r)!r!} \). This simplification shows that the denominators of the two sides are identical, which proves that the expression is valid for all integers \( r \) and \( n \) where \( 0 \le r \le n \).

Key Concepts

Understanding Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and combination. The subject delves into what is possible when you combine objects following specific rules. For example, if you're at a buffet and want to know how many different plates of food you can create from a selection of dishes, combinators would be your go-to tool. Such real-life scenarios translate into mathematical problems about combinations, which are a fundamental part of combinatorics.

Understanding the basic principles of combinatorics is essential in solving problems that ask for the number of possible arrangements or selections without going through the laborious process of listing them all out. This brings us to a critical combinatorial concept, the binomial coefficient, often expressed using the notation C_r, which represents the number of ways to choose r items from a set of n distinct items without regards to the order. A standard buffet problem in combinatorics could be reformulated into a question about binomial coefficients.

Decoding Factorial Notation

Factorial notation is a mathematical expression that represents the product of a series of descending natural numbers. It is denoted by the exclamation mark (!). In the context of combinatorics, factorials play a crucial role, particularly in formulae related to permutations and combinations.

For a positive integer n, the notation n! (read as 'n factorial') is the product of all positive integers less than or equal to n. Here’s how it looks in action:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 3! = 3 × 2 × 1 = 6
• 1! = 1 (by definition, the factorial of 1 is itself)
• 0! = 1 (by convention, the factorial of 0 is defined to be 1)

The concept of factorial is vital because it helps us count the number of ways objects can be arranged (called permutations), and, as we have seen in binomial coefficients, it is central to calculating combinations.

Properties of Combinations

When exploring the properties of combinations, we encounter various symmetries and identities that reveal the beautiful structure of binomial coefficients. One such property is the symmetry property, which indicates that choosing r elements from a set of n is equivalent to choosing the n - r elements that are left. In mathematical terms, this is expressed as C_r = _nC_{n - r}.

The underlying reason for this property is because when you choose r elements, the remaining n - r elements are automatically not chosen, and vice versa. This property is quite logical yet profound, as it simplifies computations and provides insights into the inherent balance in the combinatorial world. When approaching problems in combinatorics, recognizing these properties can both check your work and enhance your understanding of the problem at a deeper level. The symmetry property exemplifies how combinatoric principles can reveal simple solutions to seemingly complex problems.