Problem 1 - Algebra 2

Question

Evaluate each expression. \(P(5,3)\)

Step-by-Step Solution

Answer

P(5,3) = 60

Step 1: Understand Permutation Notation

The permutation notation \(P(n,r)\) represents the number of ways to arrange \(r\) objects from \(n\) objects, where order matters. In this case, we are determining \(P(5,3)\).

Step 2: Apply the Permutation Formula

The formula for permutations is \(P(n,r) = \frac{n!}{(n-r)!}\). For \(P(5,3)\), we substitute \(n=5\) and \(r=3\) into the formula.

Step 3: Calculate the Factorials

Calculate \(5!\) and \((5-3)!\):- \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)- \(2! = 2 \times 1 = 2\)

Step 4: Substitute and Solve the Expression

Using the permutation formula, we have:\[ P(5,3) = \frac{5!}{(5-3)!} = \frac{120}{2} \]

Step 5: Simplify the Expression

Divide 120 by 2 to get:\[ P(5,3) = 60 \]

Key Concepts

Understanding Factorials

Factorials are an essential concept when working with permutations, as they help us determine the number of ways to arrange a set of objects. In mathematics, a factorial is denoted by an exclamation mark, such as \( n! \). It represents the product of all positive integers up to a number \( n \).

• For instance, \( 5! \) is equal to \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
• Similarly, \( 3! \) would be \( 3 \times 2 \times 1 = 6 \).

Factorials grow rapidly with larger numbers, as they count down multiplicatively. They are crucial in calculating permutations because they provide the basis for determining how many different ways objects can be arranged.

Applying the Permutation Formula

The permutation formula is a mathematical tool used to find the number of different ways to arrange a subset of objects from a larger set. This is particularly useful when the order of objects matters. The formula for permutations is:\[ P(n,r) = \frac{n!}{(n-r)!} \]where \( n \) is the total number of objects, and \( r \) is the number of objects to be arranged. The subtraction \( n-r \) represents the number of items not chosen, and using factorials helps in calculating the total arrangements.For example, when solving \( P(5,3) \), we plug \( n=5 \) and \( r=3 \) into the formula:\[ P(5,3) = \frac{5!}{2!} \] This results in reducing the permutation calculation by simplifying the factorials accurately.

Arranging Objects with Permutations

Permutations allow us to find out how many different sequences can be created from a set of objects, considering the order of those objects. This is why the order of arrangement plays a key part in distinguishing permutations from combinations.

• For example, three letters \( A, B, \) and \( C \) can be arranged in multiple sequences like \( ABC, CAB, \) and \( BAC \).
• Each arrangement is different and counted separately, which is the essence of permutations.

In our example, \( P(5,3) = 60 \) means there are 60 distinct ways to arrange 3 objects from a total of 5, highlighting the flexibility and variety of arrangements achievable with permutations.