Problem 2 - College Algebra

Question

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{7} P_{3} $$

Step-by-Step Solution

Answer

The value of \(_{7} P_{3}\) is \(210\)

Step 1: Understanding the permutation formula

The permutation formula is represented as \(_{n} P_{r} = \frac{n!}{(n-r)!}\), where 'n' represents the total number of items and 'r' represents the number of items being chosen. In this case, 'n' is 7 and 'r' is 3.

Step 2: Substitute the values into the formula

So, \(_{7} P_{3} = \frac{7!}{(7-3)!}\)

Step 3: Compute factorial

Compute individually for \(7!\) and \((7-3)!\) which results in \(5040\) and \(24\) respectively.

Step 4: Divide

Finally, divide \(5040\) by \(24\) to get the result which is equal to \(210\).

Key Concepts

Factorial

Factorial is a mathematical operation that multiplies a number by all the positive integers below it. It's represented by an exclamation point (!) after the number. For example, the factorial of 5 is written as 5! and is calculated as 5 × 4 × 3 × 2 × 1, which equals 120.

Factorials are a key component in permutations and combinations, providing a way to calculate how many different ways items can be arranged or selected.

A few important properties of factorials include:

• 0! is always equal to 1. This might seem counterintuitive, but it is defined this way to help make various mathematical formulas work smoothly, particularly in permutations and combinations.
• Factorials grow very fast as the numbers increase. For instance, while 5! is 120, 10! is already 3,628,800.

Factorials are used in numerous places in mathematics, particularly in permutations, which we'll explore next.

Permutation Formula

The permutation formula is a method used to determine the number of ways a certain number of items can be arranged in a specific order. When it comes to permutations, the sequence in which you arrange the items is crucial.

The formula for permutations of n items taken r at a time is given by: \[ _{n}P_{r} = \frac{n!}{(n-r)!} \]

Let's break down what this means:

• 'n' is the total number of available items.
• 'r' is the number of items you want to arrange.

By using the permutation formula, you can establish how many possible ways you can order a subset of a larger set. For example, in calculating _{7}P_{3}, the formula becomes: \[ _{7}P_{3} = \frac{7!}{(7-3)!} = \frac{5040}{24} = 210 \] This means there are 210 different ways to arrange 3 items from a collection of 7 distinct items. Understanding the permutation formula is essential in combinatorics, which encompasses counting problems generally.

Combinatorics

Combinatorics is a branch of mathematics concerned with counting, arrangement, and combination of objects. It's a vast field with applications in various areas including statistical analysis, computer science, and optimization problems.

Two of the fundamental concepts in combinatorics are permutations and combinations. While permutations focus on the order of elements, combinations deal with selection where order does not matter.

• In permutations, as described above, the order of selection is important. For example, arranging books on a shelf.
• In combinations, the order of selection is not important. For instance, selecting members for a committee.

Combinatorics often utilizes the permutation formula to solve problems where the order of items is important. Understanding the difference between and appropriate application of permutations and combinations is crucial for solving various counting problems in mathematics.