Problem 7 - Algebra and Trigonometry

Question

Evaluate the expression. $$ P(11,4) $$

Step-by-Step Solution

Answer

The value of \( P(11, 4) \) is 7920.

Step 1: Understand the Problem

The problem asks us to evaluate the expression \( P(11, 4) \). This notation \( P(n, r) \) typically represents permutations, which means we need to find the number of possible arrangements of \( r \) items selected from \( n \) items.

Step 2: Identify the Permutation Formula

The formula for permutations is given by: \( P(n, r) = \frac{n!}{(n-r)!} \), where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

Step 3: Substitute Values into the Formula

Substitute \( n = 11 \) and \( r = 4 \) into the permutation formula: \[ P(11, 4) = \frac{11!}{(11-4)!} = \frac{11!}{7!} \]

Step 4: Calculate the Factorials

Calculate the factorials: 1. \( 11! = 11 \times 10 \times 9 \times 8 \times 7! \)2. \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)Substituting back, notice that \( 7! \) cancels from the denominator and from the part of \( 11! \).

Step 5: Simplify the Expression

After canceling \( 7! \), only the terms in \( 11! \) that remain in the numerator are: \[ 11 \times 10 \times 9 \times 8 \]

Step 6: Compute the Result

Now compute the product: \[ 11 \times 10 = 110 \]\[ 110 \times 9 = 990 \]\[ 990 \times 8 = 7920 \]Thus, the value of \( P(11, 4) \) is 7920.

Key Concepts

Factorial

In mathematics, the concept of factorial is fundamental, especially in the study of permutations and combinations. A factorial, denoted by an exclamation mark ( ! ), represents the product of an integer and all the integers below it. For any positive integer n , the factorial is expressed as n! . For example:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 4! = 4 × 3 × 2 × 1 = 24

Factorials grow very quickly with larger integer values, becoming useful in calculations involving permutations. When dealing with expressions like P(n, r) , understanding factorials is key to simplifying and computing the number of arrangements.

Arrangements

Arrangements refer to the different ways in which a set of items can be ordered. In the context of permutations, arrangements are specific sequences or orderings of a subset of items from a larger set. For example, if we have three letters A, B, and C, the different arrangements of these letters are ABC, ACB, BAC, BCA, CAB, and CBA.
In permutations, the concept of arrangements is central because it focuses on order. Unlike combinations, where the arrangement of selected items doesn't matter, permutations consider each unique ordering as a distinct possibility.
The practical use of arrangements helps in fields like cryptography, game design, and solving puzzles where the order of elements is crucial.

Permutation formula

The permutation formula is a key tool in combinatorial mathematics for finding the number of possible arrangements of a subset of items. Given any set, the symbol P(n, r) denotes the number of permutations of r items that can be drawn from a set of n items. The formula for this is:

• \[ P(n, r) = \frac{n!}{(n-r)!} \]

This formula highlights the role of both the total number of items ( n) and the number of items to arrange ( r). By factoring out (n - r)! from n!, we calculate how many ways to arrange the first r items. The permutation formula is foundational in the calculation of arrangements where the sequence matters.

Combinatorial mathematics

Combinatorial mathematics is a branch of mathematics concerning the study of counting, arrangement, and combination of sets of elements. It encompasses a wide range of topics including permutations, combinations, and their properties.
Permutations are a fundamental part of combinatorics because they count the various possible orders of elements in a set. Understanding concepts like factorial and permutations is crucial because it allows mathematicians and scientists to model real-world problems involving counting and arrangement.
Combinatorics is widely applied in fields such as computer science for algorithm design, genetics for understanding gene sequences, and logistics for optimizing routes and schedules. By using combinatorial techniques, complex problems can be broken down into simpler, quantifiable parts, making them easier to analyze and solve.