Problem 78 - 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.) $$C(n, n)=1$$

Step-by-Step Solution

Answer

\(C(n, n) = 1\) because there is exactly one way to select all items from a set of \(n\) items.

Step 1: Understanding Combinations

In combinatorics, a combination is a selection of items without considering the order. The formula for the number of combinations of selecting \(r\) items from \(n\) items is given by \(C(n, r) = \frac{n!}{r!(n-r)!}\).

Step 2: Identify the Given Expression

We need to prove the statement \(C(n, n) = 1\). This means selecting \(n\) items from \(n\) items, which intuitively involves selecting all the items.

Step 3: Apply the Combination Formula

Substitute \(n\) for \(r\) in the combinations formula: \[ C(n, n) = \frac{n!}{n!(n-n)!} \]. Simplifying the expression gives \[ C(n, n) = \frac{n!}{n! \cdot 0!} \].

Step 4: Simplify the Factorial Terms

From the formula, we have \(0! = 1\). Thus, \[ C(n, n) = \frac{n!}{n! \cdot 1} = \frac{n!}{n!} \].

Step 5: Conclude by Simplification

Simplifying \(\frac{n!}{n!}\) results in 1. Therefore, \(C(n, n) = 1\). This indicates that there is exactly one way to select all \(n\) items from \(n\) items.

Key Concepts

Permutations

In combinatorics, a permutation is essentially an arrangement of items where the order matters. Imagine you have a set of books and you want to know in how many different sequences you could arrange them on a shelf. That’s a permutation problem. When dealing with permutations, you use the formula\[ P(n, r) = \frac{n!}{(n-r)!} \]where:

• \(n\) is the total number of items,
• \(r\) is the number of items to arrange,
• \(!\) denotes a factorial, which we’ll cover next.

This formula calculates how many ways you can choose and order \(r\) items out of \(n\). For example, if you wanted to determine how many ways you can organize 3 books out of 5, simply plug \(5\) and \(3\) into the formula. Permutations are widely used in situations where the sequence or order is critical.

Factorials

The concept of a factorial is fundamental in both permutations and combinations. The factorial of a positive integer, denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). So,\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \]For example, \(5!\) (read as "five factorial") is calculated as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]An important aspect of factorials is understanding the convention that \(0! = 1\). This is a mathematical convenience that helps maintain consistent results in formulations, like the one seen in combinations.

• Factorials grow rapidly with increasing \(n\).

They are key to determining the number of permutations and combinations for a set of elements.

Combinations

Combinations are a way to select items from a group, where the order of selection does not matter. The formula to determine the number of combinations is given as:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Here:

• \(n\) is the total number of items,
• \(r\) is the number of items to choose,
• \(n!\) accounts for the permutation of all items, and dividing by \(r!\) and \((n-r)!\) removes the variations in order.

To prove the statement \(C(n, n) = 1\), recall that choosing all \(n\) items from \(n\) means there's only one way to select everything, leaving no choice among them. Hence, the combination formula confirms this because:\[ C(n, n) = \frac{n!}{n!(n-n)!} = \frac{n!}{n! \times 1} = 1 \]Demonstrating how combinatorial choices reflect intuitive understanding that selecting everything yields only one combination.

Positive Integers

Positive integers are a fundamental part of mathematics and are defined as all whole numbers greater than zero (e.g., 1, 2, 3, ...). They are often used to define sets wherever countable items are concerned, which includes problems involving permutations and combinations.

• They are used in calculating factorials, as factorials apply only to non-negative integers, typically positive.
• The arrangement counts in permutations and combinations frequently involve these numbers.

Positive integers ensure that all operations involved, such as forming permutations or calculating factorials, have meaningful and predictable outcomes. In the challenge of selecting or arranging objects, only positive integers can effectively quantify the items or positions in play. Hence, their role is vital in defining and solving combinatorial problems.