Problem 13 - Algebra 2

Question

Evaluate each expression. \(P(7,5)\)

Step-by-Step Solution

Answer

The value of \(P(7,5)\) is 2520.

Step 1: Understand the Permutation Formula

The formula for permutations of selecting \( r \) objects from \( n \) objects is given by:\[P(n, r) = \frac{n!}{(n-r)!}.\]In this exercise, we need to evaluate \( P(7,5) \). This means \( n = 7 \) and \( r = 5 \).

Step 2: Calculate the Factorials

Calculate the factorials involved.\( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)and \( (7-5)! = 2! = 2 \times 1 \).

Step 3: Substitute into the Formula

Substitute the calculated factorials into the permutation formula:\[P(7, 5) = \frac{7!}{2!} = \frac{5040}{2}\]

Step 4: Simplify the Expression

Divide 5040 by 2 to simplify the expression:\[P(7, 5) = 2520\]

Key Concepts

Factorials

To understand permutations, we first need to familiarize ourselves with the concept of factorials. A factorial, denoted by an exclamation point (!), represents the product of an integer and all the numbers below it. For instance, the factorial of 5 is written as 5!, and it equates to:\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]Factorials are essential in combinatorics because they help calculate the number of ways to arrange or select objects. The factorial n! gives us the total number of ways to arrange n distinct objects. In our exercise, we dealt with 7!, which means arranging seven items, calculated as:\[7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\]Shortcuts and calculators often handle the heavy lifting for large factorials, but it's helpful to know the process for small numbers.

Permutation Formula

The permutation formula is used to determine the number of ways to arrange a subset of items from a larger set. Permutations are all about the order of arrangement, which makes them different from combinations where order does not matter.The formula for finding permutations of n objects taken r at a time is:\[P(n, r) = \frac{n!}{(n-r)!}\]This formula divides the total arrangements of n objects (n!) by the arrangements of the remaining objects after choosing r (\((n-r)!\)). Consider our example, where we compute \(P(7, 5)\). It involves selecting 5 objects from a total of 7 available options, using:\[P(7, 5) = \frac{7!}{(7-5)!} = \frac{5040}{2!} = \frac{5040}{2} = 2520\]Here, we see the subtraction in the denominator simplifies the calculation by considering only the needed selections from the total arrangements.

Combinatorics

Combinatorics is a branch of mathematics that studies the counting, arrangement, and combination of objects within a set. It encompasses permutations, combinations, and several other concepts, making it a valuable tool in fields like probability and statistics.While permutations consider the order of items, combinatorics allows for both ordered and unordered analyses, which include:

• Permutations: Concerned with ordering.
• Combinations: Selecting items without regard for order.

Our exercise specifically focuses on permutations. This concept is crucial when the sequence of selection is important. Combinatorics often handles questions like "How many different ways can the letters in a word be arranged?" For instance, the word "apple" can be rearranged in various ways as a permutation problem. By applying combinatorial methods, you can solve complex problems in logical steps, as illustrated in our example with \(P(7, 5)\). This approach simplifies the counting process, aiding in efficient problem-solving.