Problem 77 - 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.) $$P(n, 0)=1$$

Step-by-Step Solution

Answer

\( P(n, 0) = 1 \) because choosing 0 items is possible in one way.

Step 1: Recall the Definition of Permutations

The number of permutations of choosing and arranging \( r \) objects out of \( n \) is given by the formula: \[ P(n, r) = \frac{n!}{(n-r)!} \]where \(!\) denotes factorial, the product of all positive integers up to that number.

Step 2: Substitute Values into the Permutation Formula

For the case \( r = 0 \), substitute into the permutations formula:\[ P(n, 0) = \frac{n!}{(n-0)!} = \frac{n!}{n!} \]

Step 3: Simplify the Expression

The expression \( \frac{n!}{n!} \) simplifies to 1, since any non-zero number divided by itself is 1. Thus, \( P(n, 0) = 1 \).

Step 4: Interpretation of the Result

The result makes sense because if we are choosing 0 objects out of any \( n \) objects, there is exactly one way to do so: choosing nothing. Hence, \( P(n, 0) = 1 \).

Key Concepts

Combinations

Combinations are an essential concept in mathematics and are used to determine how many different ways you can select a subset of items from a larger set, without considering the order of the items. Unlike permutations, where order matters, with combinations it does not. This is often useful when you care about the group as a whole compared to individual configurations.

The number of combinations for choosing \( r \) items from a set of \( n \) items is given by the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Here, the factorials help account for all possible arrangements of the items, which are then divided to ensure that the order of arrangement is not considered. This concept is particularly helpful in scenarios like selecting team members from a group or choosing ingredients for a recipe where the arrangement does not matter but the content does.

By understanding combinations, you set the foundation for tackling complex probability problems and making sense of different statistical models.

Factorial

Factorial is a fundamental mathematical operation used in permutations and combinations. It is denoted by an exclamation mark \(!\) and represents the product of all positive integers up to a given number. For a number \( n \), the factorial is

• \( n! = n \times (n-1) \times (n-2) \times \,\cdots\,\times 2 \times 1 \)

Factorials grow very quickly as \( n \) increases, making them crucial in calculating large permutations and combinations.

For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \), showing how rapidly these numbers increase. Factorials are commonly used in many areas of mathematics, including algebra, calculus, and discrete mathematics.

Knowing how factorials operate allows for simplification in solving equations and can significantly help in breaking down complex mathematical problems more easily.

Positive Integers

Positive integers are the set of all whole numbers greater than zero. They are the basic building blocks of number theory and are used extensively in counting and ordering. This set of numbers includes

• 1, 2, 3, 4, 5, and so on.

Positive integers are significant because they do not include zero or any negative numbers, which makes them straightforward when it comes to concepts of counting or ordering in mathematics.

In the context of permutations and combinations, positive integers are often used to define the size of sets or the number of items being chosen or arranged. For example, when talking about arranging \( r \) items out of \( n \), both \( r \) and \( n \) are positive integers with the condition \( r \leq n \). This ensures all calculations stay within the realm of whole numbers, keeping the solutions intuitive and aligned with practical counting scenarios.

Understanding positive integers helps in grasping the entirety of basics in mathematics and forms a stepping stone to mastering more advanced topics.