Permutation and Combination: Definitions and Formulas

Permutation and Combination are two important concepts in mathematics used to calculate arrangements and selections of items in a group.

1. Permutation

Definition:

• Permutation refers to the arrangement of items in a specific order.
• It considers the order of the items.

Formula:

P(n,r)=n!/(n−r)!

Where:

• n = Total number of items.
• r = Number of items to arrange.
• n! (n factorial) = Product of all positive integers up to nn.

Key Points:

• Order matters.
• Example: Arranging 3 letters A,B,CA, B, C in different ways gives ABC,ACB,BAC,BCA,CAB,CBAABC, ACB, BAC, BCA, CAB, CBA.

2. Combination

Definition:

• Combination refers to the selection of items without considering the order.
• The arrangement does not matter.

Formula:

C(n,r)=n!/r!(n−r)!

Where:

• n = Total number of items.
• r = Number of items to select.

Key Points:

• Order does not matter.
• Example: Selecting 2 letters from A,B,CA, B, C gives combinations AB,AC,BCAB, AC, BC.

Key Differences

Examples

Example 1: Permutation

How many ways can 3 people (A, B, C) sit in a row of 3 seats?

P(3,3)=3!/(3−3)!=3!/0!=6

Arrangements: ABC,ACB,BAC,BCA,CAB,CBAABC, ACB, BAC, BCA, CAB, CBA.

Example 2: Combination

How many ways can 2 people be selected from 3 (A, B, C)?

C(3,2)=3!/2!(3−2)!=3!/2!⋅1!=6/2 = 3

Selections: AB,AC,BCAB, AC, BC.