Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$C(6,0)$$

Step-by-Step Solution

Answer

\( C(6,0) = 1 \)

Step 1: Understanding the Combination Formula

The combination formula is used to find the number of ways to choose a subset of items from a larger set, in which the order of the items does not matter. It is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items, and \( r \) is the number of items to choose. In this problem, \( n = 6 \) and \( r = 0 \).

Step 2: Applying the Zero Combination Rule

According to the rule for combinations, choosing all 0 items from any number of items (\( n \)) results in exactly 1 way. Mathematically, this is represented by: \[ C(n, 0) = 1 \] Therefore, for our expression, \( C(6, 0) = 1 \).

Key Concepts

Factorials

Factorials are an important concept in mathematics and are denoted by an exclamation mark (!). They represent the product of all positive integers up to a given number. For example, if you want to find the factorial of 5, you calculate it as:\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]This operation is crucial when dealing with permutations and combinations, as it helps in determining the total possible arrangements of a set.

• The factorial of 0 is a special case. By definition, it is equal to 1, i.e., \(0! = 1\).
• Factorials grow rapidly with larger numbers, making them a powerful tool for counting.

Understanding how factorials work gives you insight into calculating combinations efficiently. They play a key role in the formula for combinations and permutations, providing the count of possible options.

Subset Selection

Subset selection refers to the process of choosing a smaller set of elements from a larger collection, where the order of the selection does not matter. This concept is often encountered in statistics and probability when dealing with questions like, "How many ways can we choose a few items from a group?" The formula used for subset selection through combinations is:\[C(n, r) = \frac{n!}{r! (n-r)!}\]where

• \( n \) is the total number of items, and
• \( r \) is the number of items to choose.

In our exercise example, when applying the formula with \( n = 6 \) and \( r = 0 \), you essentially calculate how many ways you can pick zero items from six. This is primarily about recognizing that the order does not matter in such selections.

Zero Combination Rule

The zero combination rule is a special rule in combinations that simplifies calculations significantly. It states that choosing zero elements from any set always results in exactly one possible combination, irrespective of the number of items in the set. Mathematically, it is expressed as:\[C(n, 0) = 1\]This is because there's only one way to select nothing from something: do not select anything at all. Some key points to remember:

• It is always true that \( C(n, 0) = 1 \), highlighting the consistent nature of this concept across different scenarios.
• This rule is efficient because it negates the need for lengthy calculations when zero items are to be selected.

Recognizing and applying the zero combination rule allows for quick solutions, especially when dealing with larger sets, where manually calculating factorials is not practical.