Problem 37 - College Algebra with Corequisite Support

Question

Can \(C(n, r)\) ever equal \(P(n, r) ?\) Explain.

Step-by-Step Solution

Answer

Yes, when \(r = 1\).

Step 1: Understanding Combinations

The expression for combinations, denoted as \(C(n, r)\), is given by \(\frac{n!}{r!(n-r)!}\). This represents the number of ways to choose \(r\) items from \(n\) items without considering the order.

Step 2: Understanding Permutations

The expression for permutations, represented as \(P(n, r)\), is given by \(\frac{n!}{(n-r)!}\). This counts the number of ways to arrange \(r\) items from a total of \(n\) items, so the order does matter.

Step 3: Analyzing When Combinations Equals Permutations

For \(C(n, r)\) to equal \(P(n, r)\), their expressions must be equivalent: \(\frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!}\). If we simplify this equation, we find that it only holds when \(r! = 1\). This implies \(r = 1\) since the factorial of 1 is 1.

Step 4: Conclusion

\(C(n, r) = P(n, r)\) when \(r = 1\). In this case, combinations and permutations are equivalent because selecting one item from \(n\) items doesn't involve multiple orders.

Key Concepts

Permutations

Permutations are an important concept in combinatorics, focusing on the arrangements of a set of items. They emphasize the order of these arrangements. For example, consider arranging letters or numbers where the position is crucial. When we talk about permutations mathematically, we use the notation \( P(n, r) \), where \( n \) is the total number of items and \( r \) is the number of items we're choosing to arrange. The formula for permutations is:\[P(n, r) = \frac{n!}{(n-r)!}\]

• Factorial Use: The factorial function, represented by \( n! \), helps calculate the permutation by reducing possibilities as slots are filled.
• Order Matters: In permutations, rearranging the same items will give a different result, unlike combinations.

Whenever you're dealing with problems where sequence plays a role—such as seating arrangements or race outcomes—permutations are your best friend.

Combinations

Combinations differ from permutations by ignoring the order of the arranged items, focusing solely on selection. This is like selecting a group of friends to go out, where the individual choice is important, but the order doesn't change the group.Combinations are denoted as \( C(n, r) \), where \( n \) is the number of available items and \( r \) is how many items we pick. The formula for combinations is:\[C(n, r) = \frac{n!}{r!(n-r)!}\]

• No Order Concern: Combinations care only about which items you take, not the sequence of selection.
• Simple Calculations for Outcomes: Useful in situations like lottery selection where the drawing order doesn't matter.

Combinations essentially allow us to count groupings without respect to arrangement.

Factorial

Factorials are a key mathematical concept used in permutations and combinations, often denoted by \( n! \), which means the product of all positive integers up to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).Factorials grow extremely fast and play a critical role in calculating permutations and combinations:

• Permutations: Factorials help determine different orderings of items by factoring down for each position.
• Combinations: Factorials adjust for the lack of order significance, dividing out repeated arrangements.

Factorials also appear in many other areas of mathematics, such as series expansions and probability, making understanding their behavior quite beneficial.