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Factorial:

Factorial is just the multiplication of n with all its preceding integer numbers till 1.
so,
1! = 1
2! = 2 × 1
3! = 3 × 2 × 1
4! = 4 × 3 × 2 × 1
and so on.

we define, 0! = 1

Permutations:

A Permutation is an arrangement in a definite order of a number of objects taken some or all at a time.
Or, Shuffling, basically.

Formula:

= n! ÷ (n – r)!

Combinations:

A Combination is a selection of items taking all or some at a time.But the order of items does not matter.For example, in combinations AB and BA is the same.

Formula:

= n! ÷ (r! × (n – r)!)

Example of permutation and combination.

In this Kotlin code, Im going to represent
as (n P r) and as (n C r). And in Console screen I'll be representing as P(n, r) and as C(n, r). As it is the same thing.

Some Theorems:

Theorem 1:
For every permutation and combination, the n is a constant positive integer, since it represent the total number of objects, and r is a integer which lies between: 0 to n, since it represent the number of objects to be selected.

Theorem 2:
The number of permutation of n objects, where s₁ objects are of one kind, s₂ objects are of second kind, ... s_k of k^th kind and the rest, if any, are of different kind is:

n! ÷ (s₁! × s₂! × ... × s_k!)

Theorem 3:
The Relationship between permutation and combination is:

= r! ×

Theorem 4:
In combinations, it was found that:

+ ⁿC_{r –1} = ^{n +1}C_r

Click Here to see its proof.

Lesson on P&C