Problem 19 - College Algebra

Question

For the following exercises, compute the value of the expression. $$ P(11,5) $$

Step-by-Step Solution

Answer

The value of \( P(11, 5) \) is 55440.

Step 1: Identify the formula for permutations

The notation \( P(n, r) \) refers to the number of permutations of \( n \) items taken \( r \) at a time. The formula for permutations is given by:\[ P(n, r) = \frac{n!}{(n-r)!} \]

Step 2: Substitute the given values into the formula

Here, \( n = 11 \) and \( r = 5 \). Substitute these values into the permutation formula:\[ P(11, 5) = \frac{11!}{(11-5)!} \]

Step 3: Simplify the expression

Calculate the factorial values. First, compute the denominator:\( (11-5)! = 6! \). Now write the expression:\[ P(11, 5) = \frac{11!}{6!} \]

Step 4: Calculate the factorials

Compute \( 11! \) and \( 6! \) using their respective expansion:\[ 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6! \]By expanding the factorials, we see the \( 6! \) in the numerator and denominator cancel each other out.

Step 5: Compute the result

The remaining expression after canceling \( 6! \) from the numerator and denominator is:\[ 11 \times 10 \times 9 \times 8 \times 7 = 55440 \].Thus, \( P(11, 5) = 55440 \).

Key Concepts

Factorials

Factorials are foundational mathematical functions used to calculate permutations and combinations, amongst other things. The notation for a factorial is an exclamation mark '!' following a number, for instance, 11!. This notation means you multiply all the natural numbers up to and including the given number. For example, 11! = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. Factorials grow very quickly as the numbers increase, hence they are a powerful mathematical tool.
When you're asked to calculate permutations like in the problem P(11,5), you're using factorials to determine how many ways you can arrange a specific number of items from a larger set. Factorials are critical because they represent possible arrangements, simplifying complex arrangements into manageable calculations.
To compute factorial manually:

• Start with the number given.
• Multiply it by each preceding integer down to 1.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns over finite sets. It includes both permutations and combinations, which help determine how items can be selected or arranged.
A permutation specifically refers to arranging items where the order is significant. Hence, in P(11,5), we ask, "In how many ways can 5 items be arranged from a group of 11?"
Combinatorial principles are extremely useful in algebra problem-solving, computer science, and statistics since they offer ways to calculate probabilities and organize data.

• Permutations: Order matters when arranging items.
• Combinations: Order does not matter when selecting items.

Combining these principles with factorials allows one to estimate the possible outcomes efficiently.

Algebra Problem-Solving

Algebra problem-solving often includes using permutations and combinations to find solutions to real-world problems or theoretical exercises. When tackling problems like P(11,5), understanding the flow of working through algebraic equations is crucial.
This involves breaking down the problem as follows:

• Identify the type of problem: permutation or combination.
• Use the correct formula: factorials are key players here.
• Substitute with given values to simplify equations.
• Perform calculations step-by-step: Start from the simplest operations like factorial estimation and work through to more complex arrangements.

Permutations, being core to algebraic problem-solving, require clear steps and logical thinking. Whether in academic exercises or real-world applications, such understanding turns abstract calculation into accessible solutions. This methodical approach keeps the anxiety of large numbers at bay and builds confidence in problem-solving skills.