Problem 4 - Algebra and Trigonometry

Question

1–6 Evaluate the expression. $$P(10,5)$$

Step-by-Step Solution

Answer

The value of \(P(10, 5)\) is 30,240.

Step 1: Understand Permutation Notation

The notation \(P(n, r)\) represents a permutation, which is used to determine the number of ways to arrange \(r\) objects out of \(n\) distinct objects. The formula for permutations is given by: \(P(n, r) = \frac{n!}{(n-r)!}\).

Step 2: Identify the Values of n and r

In the given expression \(P(10, 5)\), the value of \(n\) is 10 and the value of \(r\) is 5. This means we want to find the number of ways to arrange 5 objects out of a total of 10 objects.

Step 3: Apply the Permutation Formula

Using the formula \(P(n, r)\), calculate \(P(10, 5)\) by substituting \(n = 10\) and \(r = 5\) as follows:\[P(10, 5) = \frac{10!}{(10-5)!} = \frac{10!}{5!}\]

Step 4: Calculate Factorials

Calculate \(10!\) and \(5!\):- \(10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)- \(5! = 5 \times 4 \times 3 \times 2 \times 1\)

Step 5: Simplify and Solve

Compute \(P(10, 5)\) by dividing the expanded factorials:\[P(10, 5) = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{5!}\]The \(5!\) cancels out, so:\[P(10, 5) = 10 \times 9 \times 8 \times 7 \times 6 = 30240\]

Step 6: Conclusion

The value of the permutation \(P(10, 5)\) is 30,240. There are 30,240 different ways to arrange 5 objects out of 10 distinct objects.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, particularly useful in permutations and combinations. A factorial, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number.
For example, the factorial of 5 (written as 5!) is calculated as:

• 5! = 5 × 4 × 3 × 2 × 1 = 120

Factorials grow very quickly as the numbers increase, which makes them powerful in solving various problems related to arrangements and probability. They are particularly important in permutations, where we are interested in the different arrangements of a set of objects.
Remember, 0! is defined to be 1. This may seem unusual, but it helps in keeping formulas consistent, especially in permutations and combinations.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns in sets. It's often about figuring out how many ways certain selections or arrangements can be made.
For example, in a situation where you have to choose 3 ice cream flavors out of 10 available, combinatorics helps in determining the number of possible combinations.

• Permutations: Concerned with the arrangements of objects where order matters.
• Combinations: Deal with the selection of objects where order does not matter.

Combinatorics is crucial in fields such as computer science, cryptography, and even in everyday decision making, where you need to consider various possibilities and outcomes. By breaking down complex problems into simpler ones, combinatorics provides critical insights into complex systems and decision making processes.

Permutation Formula

The permutation formula is used to determine the number of ways to arrange a subset of objects from a set, where the order of arrangement is important.
The general permutation formula is given by:
\[ P(n, r) = \frac{n!}{(n-r)!} \]Where:

• n is the total number of objects.
• r is the number of objects to be arranged.
• n! represents the factorial of n.
• (n-r)! represents the factorial of the difference between n and r.

To illustrate, let's use our earlier exercise of \( P(10, 5) \). Here, you calculate the number of permutations of selecting and arranging 5 items from a total of 10 items. It evaluates to \( \frac{10!}{5!} \), which after simplifying gives us 30,240 ways.
This formula is particularly useful in situations where order and arrangement are crucial, such as forming passwords, arranging people in lines, or organizing tournament schedules.