Problem 76 - 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.) $$P(n, 1)=n$$

Step-by-Step Solution

Answer

The permutation \(P(n, 1)\) equals \(n\).

Step 1: Understand the Definitions

The permutation function, denoted as \(P(n, r)\), counts the number of ways to arrange \(r\) objects from \(n\) distinct objects. It is given by the formula \(P(n, r) = \frac{n!}{(n-r)!}\). Here, \(!\) denotes the factorial function.

Step 2: Apply Formula to Given Problem

We need to prove that \(P(n, 1) = n\). Using the formula for permutations, substitute \(r = 1\): \[ P(n, 1) = \frac{n!}{(n-1)!} \]

Step 3: Simplify the Expression

The expression \(\frac{n!}{(n-1)!}\) simplifies as follows:1. The factorial \(n!\) can be expressed as \(n \times (n-1)!\).2. Substitute this into the permutation formula:\[ P(n, 1) = \frac{n \times (n-1)!}{(n-1)!} \]

Step 4: Cancel Common Terms

Cancel \((n-1)!\) from the numerator and the denominator:\[ P(n, 1) = n \]

Step 5: Conclusion

We've shown that simplifying \(P(n, 1) = \frac{n!}{(n-1)!}\) results in \(n\). Thus, \(P(n, 1) = n\) is proved, confirming the statement for all positive integers \(n\).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, particularly in topics like permutations and combinations. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number. For example, the factorial of 5, written as \(5!\), equals \(5 \times 4 \times 3 \times 2 \times 1 = 120\). This sequence of multiplication is what makes factorials grow very large quite quickly.

Factorials play a crucial role in formulas for permutations and combinations because they allow us to calculate the total number of different ways in which a set of items can be arranged or selected. To understand permutations involving \(n\) items chosen \(r\) at a time, the formula \(P(n, r) = \frac{n!}{(n-r)!}\) is used. The factorial function in this formula helps manage the decreasing number of choices as selections are made.

• Useful for counting arrangements
• Essential in defining permutations and combinations
• Involves simple multiplication of consecutive numbers

Combinatorics

Combinatorics is a branch of mathematics dealing with counting and arrangement. It involves finding out the number of ways different objects can be selected or arranged from a set. This concept is foundational when learning about permutations and combinations.

Permutations are a key concept in combinatorics. They deal with arranging a set of items where the order matters. For example, if you have three books titled A, B, and C, the different orders you can arrange them are known as permutations. When only using one slot, the permutation formula simplifies nicely, as shown when proving \(P(n, 1) = n\).

• Focuses on arrangements and selections
• Highlights the importance of order in permutations
• Combinations differ by not considering order

In contrast, combinations consider selections of items where order does not matter. For example, choosing 2 books from A, B, and C gives the same result whether you pick AB or BA; these are considered the same combination. Both permutations and combinations are central to combinatorics.

Mathematical Proof

Mathematical proof is a logical process of demonstrating that a statement is true using established mathematical principles. It is a critical skill needed to validate statements or propositions, ensuring that they hold for all cases specified.

The proof for \(P(n, 1) = n\) involves using the definition of permutations and simplifying the formula via factorials. This step-by-step process is a straightforward example of proof, engaging logical reasoning to show that arranging 1 item from \(n\) items naturally results in \(n\) different arrangements.

• Uses logical series of steps
• Employs known formulas and identities
• Ensures validity for all covered cases

Through proofs, mathematical statements are verified, transforming assumptions into accepted truths within mathematics. By understanding the underpinning logic, students can strengthen their grasp of why certain mathematical principles hold true.