Problem 82 - A Graphical Approach to College Algebra

Question

Prove each statement for positive integers \(n\) and \(r,\) with \(r \leq n\) (Hint: Use the definitions of permutations and combinations.) $$C(n, n-r)=C(n, r)$$

Step-by-Step Solution

Answer

The statement is proven since the formulas for both sides are identical.

Step 1: Understanding Permutations and Combinations

In combinatorics, the combination formula calculates the number of ways to choose a subset of items from a larger set. It's given by:\[C(n, r) = \frac{n!}{r!(n-r)!}\]where \(n!\) (n factorial) represents the product of all positive integers from \(1\) to \(n\).

Step 2: Applying the Definitions

Let's apply the combination formula to both sides of the equation we need to prove: \(C(n, n-r) = C(n, r)\). Start with the left-hand side:\[C(n, n-r) = \frac{n!}{(n-r)!(n-(n-r))!} = \frac{n!}{(n-r)!r!}\].

Step 3: Evaluating Both Sides

Now compare this with the right-hand side:\[C(n, r) = \frac{n!}{r!(n-r)!}\]. Observe that this expression is identical to what we obtained for \(C(n, n-r)\).

Step 4: Concluding the Proof

Since the combination formula for both \(C(n, n-r)\) and \(C(n, r)\) evaluates to \(\frac{n!}{r!(n-r)!}\), we have shown that both expressions are indeed equal to each other.Therefore, \(C(n, n-r) = C(n, r)\) is proven for positive integers \(n\) and \(r\).

Key Concepts

Permutations

In mathematics, permutations refer to the different ways in which a set of items can be arranged or ordered. The concept of permutation is handy when the order of selection matters.
For example, if we have three different fruits like an apple, a banana, and a cherry, there are several arrangements possible, such as apple-banana-cherry, banana-cherry-apple, and so on.
When calculating permutations, we use the term "permutation of n items taken r at a time" and it's denoted by:

• The formula: \( P(n, r) = \frac{n!}{(n-r)!} \)
• Where \( n! \) represents the factorial of n, which is the product of all positive integers up to n.

This formula helps us find how many different ways we can arrange r items from a set of n.
Permutations are crucial in fields such as cryptography and statistics where specific arrangements of data or objects can lead to different outcomes or solutions.

Combinations

Combinations are a fundamental concept in combinatorics. Unlike permutations, combinations consider the selection of elements regardless of order. This means that the combination of an apple, banana, and cherry is the same as cherry, banana, and apple.
The combination formula helps us calculate the number of ways we can select r items from a set of n items without regard to their order:

• The formula is: \( C(n, r) = \frac{n!}{r!(n-r)!} \)
• Here, \( n! \) is the factorial of n, and \( r! \) helps adjust for the duplicates that arise from the lack of concern for order.
• This results in fewer combinations than permutations for the same sets and selections.

For example, choosing 2 fruits out of 3 (apple, banana, cherry) yielding pairs like apple-banana and banana-cherry demonstrates the application of combinations.
Combinations are extensively used in probability and statistics, as well as in scenarios like lottery drawings and team selections in sports.

Factorials

Factorials are a key concept underpinning both permutations and combinations. The factorial of a number n, denoted as \( n! \), is the product of all positive integers from 1 to n. It forms a fundamental role in many areas of mathematics, particularly in combinatorics.
For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 0! \) is defined to be 1, which is a unique case and important for maintaining consistency in mathematical equations.

Factorials help simplify the expression and calculation of large numbers, especially in determining permutations and combinations where n! is a recurring element.
They also appear in mathematical areas such as algebra, calculus, and probability theory, demonstrating their versatility and importance. Understanding how factorials function is essential for grasping the core concepts of permutations and combinations, as they provide the mathematical basis for these calculations.