Problem 43 - Algebra for College Students

Question

Evaluate expression. \(P(52,0)\)

Step-by-Step Solution

Answer

1

Step 1: Understand the Permutation Formula

The permutation formula is given by \[ P(n, k) = \frac{n!}{(n - k)!} \]where \(n\) is the total number of items, \(k\) is the number of items to choose from, and \(!\) denotes factorial.

Step 2: Substitute Values into the Formula

For the given problem \(P(52, 0)\), substitute \(n = 52\) and \(k = 0\) into the permutation formula:

Step 3: Evaluate the Factorials

Calculate the factorial values needed: \[ P(52, 0) = \frac{52!}{(52 - 0)!} = \frac{52!}{52!} \]

Step 4: Simplify the Expression

Since the factorial expressions \(52!\) in the numerator and denominator are the same, they cancel each other out. Therefore, \[ \frac{52!}{52!} = 1 \]

Step 5: Conclude the Result

Hence \[ P(52, 0) = 1 \]

Key Concepts

Factorial

In mathematics, the factorial of a non-negative integer is the product of all positive integers less than or equal to that number. It's denoted by the symbol \(!\). For example, \(!5\) (read as 'five factorial') is calculated as follows:
\[ 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 \] The factorial function grows very quickly. Even for relatively small numbers, the factorial can be quite large. For 0, it's defined that \(!0 = 1\). This definition helps in simplifying various mathematical expressions and maintaining consistency in equations.

Permutation Formula

Permutations refer to the different possible arrangements of a set of items. The order in which we arrange the items matters in permutations. The formula for permutations is given by:
\[ P(n, k) = \frac{n!}{(n - k)!} \] Here, \(n\) represents the total number of items, and \(k\) is the number of items to be arranged. The factorial in the numerator (\(n!\)) provides the number of ways to arrange all \(n\) items, while the factorial in the denominator (\((n - k)!\)) removes the excess arrangements when fewer than \(n\) items are being selected. Using this formula, the specific permutation for \(P(52, 0)\) was found by substituting the values into it.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting. It provides tools and formulas to count the ways objects can be chosen or arranged. In this exercise, we used a specific formula from combinatorics to determine permutations. Combinatorics includes:

• Permutations: Arrangements where order matters.
• Combinations: Selections where order does not matter.

For permutation problems like \(P(52, 0)\), combinatorics helps in calculating the ordered arrangements efficiently through well-defined formulas.

Algebra

Algebra often deals with symbols and the rules for manipulating these symbols to solve equations or perform calculations. In solving permutation problems like \(P(52, 0)\), understanding algebraic manipulation is key. We used algebra to:

• Substitute values into a given formula.
• Simplify expressions.
• Solve factorial equations.

By applying algebraic principles, we simplify complex expressions and make sure that calculations like \(\frac{52!}{52!}\) reduce correctly to 1, ensuring the integrity of our solution.