Problem 73 - College Algebra

Question

Prove that $$ \left(\begin{array}{l}n \\\r\end{array}\right)=\left(\begin{array}{c}n \\\n-r\end{array}\right) $$

Step-by-Step Solution

Answer

The equality \(\binom{n}{r} = \binom{n}{n-r}\) is proven by comparing these two binomial coefficients. After using the binomial coefficient formula for both sides, it's clear that the two expressions are equal, which confirms the equality.

Step 1: Define binomial coefficient

The formula for a binomial coefficient is given by \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\] where \(n!\) is the factorial of n, \(r!\) the factorial of r, and \((n-r)!\) the factorial of \(n-r\). The factorial of a number is the product of all positive integers less than or equal to that number.

Step 2: Calculate binomial coefficient for \(\binom{n}{n-r}\)

Apply the same formula to calculate the binomial coefficient \(\binom{n}{n-r}\). Using the formula, \[\binom{n}{n-r} = \frac{n!}{(n-r)!(n-(n-r))!}\] Simplifying the denominator, \[\binom{n}{n-r} = \frac{n!}{(n-r)!r!}\]

Step 3: Comparison

Now comparing the two expressions \(\binom{n}{r} = \binom{n}{n-r}\), it can be seen that they are equal, as the right side equals \(\frac{n!}{r!(n-r)!}\) which is equal to the left side. This confirms the equality.

Key Concepts

Understanding Factorials

Factorials are a fundamental concept in mathematics, specifically in the areas of algebra and combinatorics. A factorial, denoted by the symbol "!", represents the product of all positive integers up to a specified number. For instance, the factorial of a number \(n\), written as \(n!\), is calculated as \(n \times (n-1) \times (n-2) \times \ldots \times 1\).
A few examples to illustrate:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(0! = 1\) (by definition, to ensure correctness in combinations and permutations)

Factorials are extensively used in the calculation of permutations and combinations, making them an integral part of any study involving counting principles or probability.

Combinatorics and Binomial Coefficients

Combinatorics is the branch of mathematics that deals with counting, arranging, and finding patterns among a set of items. A key element of combinatorics is understanding combinations, which are selections of items from a larger set without regard to the order of selection.
The binomial coefficient, denoted as \(\binom{n}{r}\), is a mathematical way to express the number of possible combinations of \(r\) items from a set of \(n\) distinct items. Its formula is given by:

• \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]

This formula uses the concept of factorials to determine how many ways you can choose \(r\) items from \(n\) items. For example, \(\binom{5}{2}\) calculates the combinations of choosing 2 items from 5, resulting in 10 possible combinations.

Algebraic Proof of Binomial Coefficient Equality

An algebraic proof is a logical process where we use algebraic expressions and identities to demonstrate the truth of a mathematical statement. In this exercise, the goal is to prove the equality \(\binom{n}{r} = \binom{n}{n-r}\).
We start by expressing both sides using the binomial coefficient formula:

• For \(\binom{n}{r}\): \[\frac{n!}{r!(n-r)!}\]
• For \(\binom{n}{n-r}\): \[\frac{n!}{(n-r)!r!}\]

Notice that both expressions simplify to the same form because the terms \(r!\) and \((n-r)!\) can be interchanged.
This equality demonstrates a fascinating property of combinations: the count of possible selections of \(r\) items from \(n\) is identical to the count of choosing \(n-r\) items, acknowledging that selecting items to involve those not selected as well.