Problem 8 - Algebra and Trigonometry

Question

Find the value of each permutation. $$ P(7,2) $$

Step-by-Step Solution

Answer

The value of P(7, 2) is 42.

Step 1: Define the Permutation Formula

Permutations of n objects taken r at a time, denoted as \(P(n, r)\), can be calculated using the formula: \[ P(n, r) = \frac{n!}{(n - r)!} \]

Step 2: Substitute Values

In the given problem, we have \(n = 7\) and \(r = 2\). Substitute these values into the formula: \[ P(7, 2) = \frac{7!}{(7 - 2)!} \]

Step 3: Simplify the Expression

First, calculate \(7 - 2\): \[ 7 - 2 = 5 \] Now our formula becomes: \[ P(7, 2) = \frac{7!}{5!} \]

Step 4: Calculate Factorials

Compute \(7!\) and \(5!\): \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Step 5: Divide the Factorials

Now, divide \(7!\) by \(5!\): \[ P(7, 2) = \frac{5040}{120} = 42 \]

Key Concepts

Permutation Formula

In combinatorics, permutations play a crucial role. A permutation of a set involves arranging all or part of the elements into a sequence or order. The permutation formula is very useful when you need to determine the number of ways to order a subset of items. The formula for permutations of n objects taken r at a time, denoted as \(P(n, r)\), is defined as:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
Here, \(n!\) (n factorial) represents the product of all positive integers up to n, and \((n - r)!\) is the factorial of the difference between n and r. This formula accounts for all possible orderings of r objects selected from a total of n. Understanding this formula aids in solving various problems in probability and statistics, making it a fundamental concept in mathematics.

Factorials

Factorials are a cornerstone concept in combinatorics and permutations. The factorial of a non-negative integer n, denoted \(n!\), is the product of all positive integers less than or equal to n. For example,
\(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\).
Factorials grow very quickly with increasing values of n. This rapid growth is why the permutation formula uses the ratio of two factorials to compute the number of arrangements efficiently. Factorials simplify complex combinatorial expressions, enabling us to solve exercises like \(P(7, 2)\) straightforwardly. In our example:
\( P(7, 2) = \frac{7!}{5!} \), where 5 is derived from \(7 - 2\). The calculation proceeds by finding the values of \(7!\) and \(5!\), making it easy to divide and simplify. Remember, factorials help handle larger calculations by breaking them down into manageable parts.

Combinatorics

Combinatorics is the field of mathematics concerning the study of finite or countable discrete structures. It encompasses a wide range of topics, including permutations, combinations, and the principals of counting. In the context of our exercise:
- **Permutations** involve arranging elements in a specific order.
- **Combinations** involve selecting elements without considering the order.
In combinatorics, understanding how to efficiently count these arrangements is key. The permutation formula lets us determine the number of ways to pick and arrange r objects from a set of n, as we did with \(P(7, 2)\). This method is not only handy for mathematics problems but also has practical applications in fields like computer science, statistics, and various engineering disciplines. By grasping the permutation formula and the concept of factorials, students can better tackle a broad array of combinatorial problems.