Permutations
- Order matters
- More number of patterns
Definition: Arrangement in a definite order of a number of objects taken some or all at a time.
ⁿPᵣ = n! / (n-r)!
where 0 ≤ r ≤ n
Examples:
- ⁸P₅ = 8 x 7 x 6
- ⁸P₃ = 8 x 7 x 6 x 5 x 4
Properties
- ⁿP₀ = 1
- ⁿPₙ = n!
Permutations with identical elements
Theorem: Number of permutations of n objects where p₁ objects are of one kind, p₂ objects are of second kind, …, pₖ objects are of kᵗʰ kind and the rest, if any are of different kinds:
n! / (p₁! * p₂! * … * pₖ!)
Uses
- Number of ways in which the letters of a word can be rearranged.
Combinations
- Order is irrelevant
- Less number of patterns
ⁿCᵣ = n! / [(n-r)! * r!]
Properties
- ⁿCᵣ = ⁿCᵣ / r!
- ⁿ⁺¹Cᵣ = ⁿCᵣ + ⁿCᵣ₋₁
- ⁿCᵣ = ⁿCₙ₋ᵣ
Uses
- Number of ways in which a deck of cards may be delt.
Result
- Number of possible solutions to Latin squares.
- Number of possible solutions sudoku.
References
- Chapter 7, NCERT class 11 mathematics textbook