The various ways in which objects or entities can be placed, set or arranged to be later selected under given conditions is called permutation and combination. Permutation and combination represent different ways to arrange discrete data and select from that particular arrangement, without replacement. The arrangement can vary from repetitive patterns to unique arrangements.
In mathematical terms, permutations are the ways of selection and combinations are the ways of arrangement.
The order of arrangement is called a permutation. It can either have a repetitive pattern or a unique way of arrangement. Order of selection is to be considered in permutation. For example, suppose 3 are to be placed on a shelf. Then the number of ways of placing the 3 books can be calculated by the formula
ⁿPᵣ = n!/(n – r)!
n is the total number of books and r is the number of books to be replaced.
Here, P is the symbol of permutation. The number of possible ways of placing 3 books in a shelf will be
³P₃ = 3!/(3 – 3)!
³P₃ = 3!/0!
³P₃ = 3 . 2
³P₃ = 6 ways
And so on, depending on the number of sides and interior angles. In Geometry circle is considered a polygon that has infinite sides.
The various ways of selection come under combination. The combination can be defined as the method of selecting objects from a given set or combination. It is unaffected by the sequence or order of selection. For example, there are 5 books on a shelf and 3 are to be chosen at random. Then the possible ways of choosing 3 books out of 5 can be calculated by the formula
ⁿCᵣ = n!/[r! . (n-r)!]
where n is the total number of books i.e. 5 i and r is the selected number of books i.e. 3. Here, C represents combination.
The possible number of ways to select 3 books out of 5 is
⁵C₃ = 5!/[3! . (5-3)!]
⁵C₃ = 5!/[3! . 2!]
⁵C₃ = 10 ways
Factorial notation is the way of representing the permutations and combinations. The sign ‘!’ is known as the factorial sign. It is used in solving questions related to binomial expansion and permutation and combination.
Factorial, which is denoted by ‘!’, can be defined as the product of all the natural numbers less than and equal to the number given. For example, the factorial of a number n is defined as the product of n and all the natural numbers less than n till 1. Symbolically, factorial n is equal to n!
n! = n . (n-1) . (n-2) . (n-3) . (n-4) ….. 4 . 3 . 2 . 1
For example, n=7, then factorial 7 is
7! = 7 . 6 . 5 . 4 . 3 . 2 . 1
7! = 5040
Points to remember