Permutation and Combination

Permutation and Combination are fundamental concepts in combinatorics used to count the number of ways objects can be selected or arranged.

1. Factorial Notation

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.
Formula: n! = n × (n-1) × (n-2) × ... × 1.
Note: 0! = 1.

2. Permutations (Arrangements)

Permutation is the arrangement of items where the order matters.

• Linear Arrangement (Distinct Objects): The number of ways to arrange n distinct objects in a row is n!. The number of ways to arrange r objects from n is given by:
  nPr = n! / (n - r)!
• Arrangements with Repeated Objects: If there are n objects where p are of one kind, q of another, and r of another, the number of arrangements is:
  n! / (p!q!r!)
• Circular Arrangement: When objects are arranged in a circle, one position is fixed. The number of ways to arrange n distinct objects in a circle is:
  (n - 1)!

3. Combinations (Selections)

Combination is the selection of items where the order does NOT matter. The number of ways to select r objects from n distinct objects is given by:
nCr = n! / [r!(n - r)!]

4. Key Differences

Use Permutation when tasks involve: Arranging letters in a word, seating people in specific chairs, or assigning specific titles (President, Secretary).
Use Combination when tasks involve: Selecting a committee, choosing fruits from a basket, or forming a hand of cards.