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Mathematical Concepts
Factorial
A factorial, denoted by n!, is the product of all positive integers less than or equal to n.
Formula: n! = n × (n-1) × (n-2) × ... × 1
Examples:
- 3! = 3 × 2 × 1 = 6
- 4! = 4 × 3 × 2 × 1 = 24
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
- 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
Permutations
A permutation is an arrangement of objects in a specific order.
Formula: P(n, r) = n! / (n-r)!
Example:
The number of ways to arrange 3 out of 5 objects: P(5, 3) = 5! / (5-3)! = 60
Combinations
A combination is a selection of objects without regard to the order.
Formula: C(n, r) = n! / [r!(n-r)!]
Example:
The number of ways to choose 3 out of 5 objects: C(5, 3) = 10
Permutations with Repetition
When repetition is allowed, the number of permutations is n^r.
Example:
The number of 3-digit numbers using digits 0-9: 10^3 = 1000
Permutations without Repetition
When repetition is not allowed, the number of permutations is P(n, r) = n! / (n-r)!.
Rearranging Letters
To find the number of ways to rearrange letters in a word, use the factorial of the number of letters.
Example:
The word "ABC" has 3 letters: 3! = 6
Circular Permutations
In circular permutations, the arrangement is in a circle, and rotations are considered the same.
Formula: (n-1)!
Example:
Arranging 5 people in a circle: (5-1)! = 4! = 24
Clockwise and Anticlockwise:
If clockwise and anticlockwise arrangements are considered different, the formula remains (n-1)!.
If they are considered the same, the formula is (n-1)!/2.
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