Permutation and Combination Lesson: Types, Formulas, and Applications

Lesson Overview

• What Is Factorial Notation?
• What Is Permutation?
• Formula for Permutations:
• Examples of Permutations:
• What Is Combination?

Permutation and Combination are two important concepts in mathematics used to calculate possible arrangements and selections of objects. They are widely used in fields like probability, statistics, and combinatorics.

Property	Permutation	Combination
Definition	Arrangement of objects in a specific order.	Selection of objects without regard to order.
Order of Elements	Order matters.	Order does not matter.
Formula	P(n,r) = n! / (n-r)!	C(n,r) = n! / (r! * (n-r)!)
Example	Arranging 3 students in a line from a group of 5.	Choosing 3 students from a group of 5 to form a team.
Use Case	Useful for arranging, ranking, or scheduling items.	Useful for selecting items or forming groups without ranking.
Result for same n and r	Larger result, as different orders count separately.	Smaller result, as order doesn't create new combinations.
Example Calculation	For 5 items taken 3 at a time: P(5,3) = 5! / 2! = 60	For 5 items taken 3 at a time: C(5,3) = 5! / (3! * 2!) = 10

The symbol (n!) used is called factorial notation. Let's quickly understand it before we discuss permutations and combinations.

What Is Factorial Notation?

Factorial notation is a mathematical operation that represents the product of all positive integers from 1 up to a given number n. It is denoted by the symbol n! and is used in various areas of mathematics, such as permutations, combinations, and probability.

Definition:

• n! (read as "n factorial") is the product of all integers from n down to 1.
• The factorial of a number n is calculated as:

n! = n×(n−1)×(n−2)×⋯×2×1

For example: 

5! = 5x4x3x2x1 = 120

What Is Permutation?

A permutation is an arrangement of objects in a specific order. In permutations, the order of elements is important-changing the order creates a different arrangement. This concept is used in situations where the sequence of items matters, such as seating arrangements, rankings, or scheduling.

Key Points of Permutation:

• Order Matters: Permutations focus on arranging objects in a particular order, so every different order counts as a unique permutation.

No Replacement: Generally, permutations assume that each item can be used only once in an arrangement.

Formula for Permutations:

To find the number of ways to arrange rrr objects out of a total of n distinct objects, the formula is:

where:

• n! (n factorial) is the product of all positive integers from n down to 1.
• (n−r)! is the factorial of the difference between the total items n and the chosen items r.

Examples of Permutations:

1. Arranging People in a Line:
   Suppose you have 3 people (A, B, and C) and want to know the different ways to arrange them in a line.

• Possible arrangements (permutations) are: ABC, ACB, BAC, BCA, CAB, CBA.

• Total permutations = 6.
  

2. Formula Example:

• If you have 5 books and want to select and arrange 3 of them on a shelf, use the permutation formula:

So, there are 60 possible ways to arrange 3 books out of 5.

When to Use Permutations:

Use permutations when you need to arrange items in order and every unique arrangement counts. Examples include:

• Ranking positions in a race (1st, 2nd, 3rd place).
• Arranging books on a shelf.
• Setting up a password where the sequence of characters matters.

Example 1:

Suppose you have 5 different books, and you want to arrange 3 of them on a shelf. In how many different ways can you arrange these 3 books?

Step-by-Step Solution

1. Identify the total number of items (n) and the number of items to arrange (r):
   • Here, the total number of books n=5.
   • We want to arrange r=3.

2. Determine if order matters:
   • Since we're arranging books on a shelf, the order does matter (changing the order of books creates a different arrangement).
   • This confirms that it's a permutation problem.

3. Use the permutation formula:

Substituting in the values n=5 and r=3:

4. Calculate 5! and 2!:
   • 5! = 5x4x3x2x1 = 120
   • 2! = 2x1 = 2

5. Plug in the values and solve:

6. Interpret the result:

There are 60 different ways to arrange 3 books out of 5 on the shelf.

Using all the letters of the word GIFT how many distinct words can be formed?

Example 2:To find the number of distinct words that can be formed using all the letters of the word "GIFT," we'll treat this as a permutation problem since the order of the letters matters, and each letter is unique.

Step-by-Step Solution:

1. Count the letters in the word "GIFT":

• "GIFT" has 4 unique letters (G, I, F, T).
  

2. Use the permutation formula for arranging all letters: Since all letters are unique, the number of distinct arrangements (permutations) of 4 letters is given by:

P (4) = 4! = 4x3x2x1 = 24

So, 24 distinct words can be formed using all the letters of the word "GIFT."

What Is Combination?

A combination is a selection of items from a larger set, where the order of selection does not matter. In combinations, different arrangements of the same items are considered the same combination. This concept is used in situations where you want to choose items or form groups, regardless of the sequence in which they are chosen.

Key Points of Combination:

• Order Does Not Matter: Unlike permutations, combinations focus solely on which items are selected, not the order they're selected in.

• No Replacement: Generally, combinations assume each item can only be chosen once.
  

Formula for Combinations:

To find the number of ways to choose r items from a total of n items, use the combination formula:

where:

• n! (n factorial) is the product of all positive integers from n down to 1.
• r! is the factorial of the number of items chosen.
• (n−r)! is the factorial of the difference between the total items n and the chosen items r.

Examples of Combinations:

Choosing Members for a Team:

Suppose you have 5 people (A, B, C, D, and E) and want to select 3 of them for a team.

• Possible combinations (ignoring order) are: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, and CDE.
• Total combinations = 10.

Formula Example:

If you have 8 flowers and want to choose 4 of them for a bouquet, use the combination formula:

1. So, there are 70 possible ways to choose 4 flowers out of 8.

When to Use Combinations:

Use combinations when the order of items does not matter, and you simply want to count the number of possible groups or selections. Examples include:

• Forming a committee from a larger group.
• Choosing a set of lottery numbers (the order in which they're drawn doesn't matter).
• Selecting toppings for a pizza (the order of toppings is irrelevant).

Example:

Suppose you have 7 people, and you want to form a committee of 3 people. In how many ways can you choose 3 people from this group of 7?

Step-by-Step Solution

1. Identify the total number of items (n) and the number of items to choose (r):
   • Here, the total number of people n = 7.
   • We want to choose r =3 people.
2. Determine if order matters:
   • Since we are forming a committee, the order does not matter (choosing people A, B, and C is the same as choosing B, C, and A).
   • This confirms that it's a combination problem.

3. Use the combination formula:

Substituting in the values n =7 and r = 3:

4. Calculate 7!, 3! and 4!:
   • 7! = 7×6×5×4×3×2×1 = 5040
   • 3! = 3×2×1 = 6
   • 4! = 4×3×2×1 = 24
     

5. Plug in the values and solve:

6. Interpret the result:

There are 35 different ways to select 3 people from a group of 7 for the committee.

Answer:

So, the correct answer is 35 combinations.