Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, n-1)=P(n, n)$$

Step-by-Step Solution

Answer

Both permutations equal \(n!\), so \(P(n, n-1) = P(n, n)\).

Step 1: Understand the Definitions

Permutations \(P(n, r)\) represent the number of ways to arrange \(r\) objects out of \(n\) distinct objects. The formula is \(P(n, r) = \frac{n!}{(n-r)!}\). In the problem, we need to prove \(P(n, n-1) = P(n, n)\).

Step 2: Calculate \(P(n, n-1)\)

Using the permutation formula \(P(n, r)\):\[P(n, n-1) = \frac{n!}{(n-(n-1))!} = \frac{n!}{1!} = n!\] This represents the number of ways to arrange \(n-1\) items out of \(n\).

Step 3: Calculate \(P(n, n)\)

Using the same permutation formula:\[P(n, n) = \frac{n!}{(n-n)!} = \frac{n!}{0!} = n!\] Since \(0!\) is defined to be \(1\), we also get \(n!\) for \(P(n, n)\).

Step 4: Compare the Results

Since both \(P(n, n-1)\) and \(P(n, n)\) result in \(n!\), they are equal. Thus, \(P(n, n-1) = P(n, n)\).

Key Concepts

Positive Integers

Positive integers are the set of whole numbers greater than zero. These are the numbers we use in counting and ordering, and they play a significant role in mathematics. They are represented as \(\{1, 2, 3, 4, \ldots\}\). In the context of permutations and combinations, positive integers specify the number of items or people that can be arranged or selected.

• In our exercise, \(n\) and \(r\) are both positive integers, representing the total items and the selected items, respectively.
• It is essential that \(0 < r \leq n\), ensuring that \(r\) is never greater than \(n\), which would be nonsensical in permutations and combinations.

Recognizing positive integers helps in defining the domain of problems and ensuring solutions are viable and logical.

Factorial

The factorial of a positive integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials are foundational in permutations and combinations, as they help in calculating possible arrangements.

• It is important to note that \(0! = 1\), which is a necessary convention for many combinatorial formulas to work correctly. In fact, this principle is used in finding permutations when \(r = n\).
• In permutations, \(P(n, r) = \frac{n!}{(n-r)!}\), factorials help in understanding how to calculate the number of ways to arrange items.

Factorials grow very fast with increasing \(n\), which is why they are significant in understanding the complexity of permutations and combinations.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of elements within a set. It explores the different ways of selecting and arranging subsets of a set.

• In permutations, order matters, and it counts how many different ways you can order a set number of items. For example, arranging 3 books from a shelf of 5.
• The formula for permutations, \(P(n, r) = \frac{n!}{(n-r)!}\), stems from combinatorial principles, helping us understand how different selections can be made.
• In this exercise, combinatorics aids in proving the equality of two specific permutations by showcasing that both arrangements result in the same number of ways to organize the items.

Understanding combinatorics provides powerful tools to solve complex counting problems, helping to derive various results in permutations and combinations.