Problem 80 - 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.) $$C(n, 1)=n$$

Step-by-Step Solution

Answer

The proof shows that \(C(n, 1) = n\).

Step 1: Understand the Combination Formula

The combination formula for selecting \(r\) items from \(n\) items is given by: \[C(n, r) = \frac{n!}{r!(n-r)!}\]where \(n!\) is the factorial of \(n\), calculated as \(n \times (n-1) \times \ldots \times 1\).

Step 2: Plug in Values for r=1

Given \(C(n, 1)\), we substitute \(r = 1\) into the combination formula:\[C(n, 1) = \frac{n!}{1!(n-1)!}\]

Step 3: Simplify the Expression

Calculate the factorial terms: \(1! = 1\) and \((n-1)!\) is the factorial of \(n-1\). Thus, the equation becomes:\[C(n, 1) = \frac{n!}{1 \times (n-1)!}\]We can express \(n!\) as \(n \times (n-1)!)\). Therefore, \[C(n, 1) = \frac{n \times (n-1)!}{(n-1)!} = n\]

Step 4: Conclude the Proof

The simplification shows that selecting 1 item from \(n\) items can be done in \(n\) different ways, hence proving:\[C(n, 1) = n\]

Key Concepts

Permutations

Permutations are arrangements of objects where the order is important. Imagine you have a set of different books on a shelf. If you move any of them, the order changes, creating a new permutation. Permutations are used when you want to figure out how many ways you can arrange a set of items.

• For instance, if you have 3 objects: A, B, and C, all possible permutations are ABC, ACB, BAC, BCA, CAB, and CBA.
• Notice that changing the order gives a new arrangement, which is different from combinations where order does not matter.

Permutations are calculated using the formula:\[P(n, r) = \frac{n!}{(n-r)!}\]Here, \(n\) is the total number of items, and \(r\) is the number of items being chosen. For example, arranging 3 out of 5 books would require finding \(P(5, 3)\). This idea is foundational in understanding other concepts such as combinations.

Factorial

Factorials are key to understanding both permutations and combinations. The factorial of a number \(n\), denoted \(n!\), is the product of all positive integers less than or equal to \(n\).
Consider \(n!\) as a building block for many mathematical calculations. It is defined as:

• \(n! = n \times (n-1) \times (n-2) \ldots \times 1\)
• For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• The special case \(0!\) is defined to be 1.

This is important because factorials simplify many expressions in algebra, especially those involving permutations or combinations. Anytime you see \(n!\), think of the multitude of sequences and sets it represents, forming the backbone of combinatorial mathematics.

Algebraic Proof

Algebraic proofs involve showing that statements are true using algebraic manipulations and logic. In this exercise, the goal is to prove the equation \(C(n, 1) = n\) for combinations.
Here's how it's done:

• Start with the combination formula: \(C(n, r) = \frac{n!}{r!(n-r)!}\).
• By plugging in \(r = 1\), we have \(C(n, 1) = \frac{n!}{1!(n-1)!}\).
• Since \(1! = 1\), the expression simplifies to \(\frac{n!}{(n-1)!}\).
• Recognizing that \(n! = n \times (n-1)!\), allows further simplification to \(\frac{n \times (n-1)!}{(n-1)!} = n\).

Thus, we've algebraically shown that selecting 1 item from \(n\) items can be done in \(n\) ways. This method highlights the elegance of using algebraic reasoning to confirm mathematical truths.