Problem 76 - Algebra 2

Question

Evaluate each expression. \(_{7} \mathrm{P}_{3}\)

Step-by-Step Solution

Answer

The number of permutations is 210

Step 1: Identify the values of n and r

In \(_{7}P_{3}\), 7 is the total number of items (n) and 3 is the number of items to choose (r). So, n = 7 and r = 3.

Step 2: Compute the factorials

Now, we will calculate factorials for both n and (n-r). In other words, we need to find the values of 7! and (7-3)!. \n 7! = 7*6*5*4*3*2*1 = 5040\n (7-3)! = 4! = 4*3*2*1 = 24

Step 3: Substitute the values into the permutation formula

We substitute the values of n! and (n-r)! into the permutation formula \( P(n,r) = \frac{n!}{(n-r)!} \). Plugging in the values we get: \( P(7,3) = \frac{5040}{24}\)

Step 4: Simplify the equation

Finally, simplify the fraction to find the total number of permutations. \( P(7,3) = \frac{5040}{24}\) = 210

Key Concepts

Factorial

When exploring permutations, a core concept utilized is the factorial, typically represented by the symbol '!'. The factorial of a number is the product of all positive integers up to that number. For instance, for the number 7, the factorial (written as 7!) is calculated as follows:

• Begin with the number 7.
• Multiply 7 by 6, then continue multiplying sequentially by every integer below it: 5, 4, 3, 2, and 1.
• The result is 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.

Factorials are crucial in the calculation of permutations because they help determine how items can be arranged in different orders. In permutations, you'll often compute not just a straightforward factorial (like 7!), but also the factorial of differences, such as (7-3)!, which simplifies permutations by accounting for selected arrangement sizes.

Permutation formula

The permutation formula is a standard tool in combinatorics used to find the number of possible arrangements of a subset of items from a larger set. It's written as \( P(n, r) = \frac{n!}{(n-r)!} \). Here:

• \(n\) is the total number of items.
• \(r\) is the number of items to arrange.
• \(n!\) refers to the factorial of \(n\).
• \((n-r)!\) represents the factorial of the difference between the total number of items and the number selected.

For example, if you have 7 items and want to choose 3, you’d use \(_{7}P_{3}\) to denote the permutation. Applying the formula: \[ P(7, 3) = \frac{7!}{(7-3)!} = \frac{5040}{24} = 210 \]This result means there are 210 different ways to arrange 3 items out of a total of 7.

Algebra 2

In Algebra 2, the concept of permutations is further explored and applied to solve complex problems involving the arrangement of items. This subject often combines various mathematical concepts, such as factorials and permutation formulas, within larger algebraic systems.
Students might encounter permutation problems in topics like:

• Combinatorics, where choosing order of items matters.
• Probability, determining likelihood of specific arrangements.
• Statistics, analyzing data permutations and their effects on results.

In Algebra 2, understanding the permutation formula allows students to translate real-world situations into mathematical models. By doing so, students can calculate probabilities and analyze outcomes efficiently. Tools like factorials and formulas provide a structured method to support learning and solve problems systematically.