Fundamental Principle of Counting (F.P.C.): If an event can occur in m different ways, following which another event can occur in n different ways, following which another event can occur in r different ways and so forth, then the total number of different ways of occurrence of the events in the given order is m x n x r x …

Note : The principle is also known as multiplication principle.

Factorial Notation:
Important Definition : Factorial is a convenient notation for representing the product of first natural numbers. The notation for the product uses a mark of exclamation (!) suffixed to the highest natural number in the product. Thus

4! = 4 × 3 × 2 × 1.
6! = 6 × 5 × 4 × 3 × 2 × 1.
In general, n! = n(n – l)(ra – 2)… (3)(2)(1).
It is read as ‘factorial n

Important formula:
(i) 6(5!) = 6(5 × 4 × 3 × 2 × 1)
= 6!
n[(n-l)!] = n!
(it) 0! = 1. [By definition]

Permutations:
Important Definition : A permutation is an arrangement in definite order of a number of objects taken when some or all at a time.

Important Formulae : (i) The number of permutations of n different objects taken all at a time, denoted by nPn is given by ⁿP_n = n(n – l)(n – 2)… 3.2.1
= n!

(ii) The number of permutations of n different objects taken r at a time denoted by ⁿP_r (n > r) is given by
ⁿP_r or P(n, r) = n(n – 1)(n – 2)… (n + 1 – r).
or ⁿP_r = \(\frac{n !}{(n-r) !}\) (when all objects are different)

(iii) Value of 0!
Putting r = n in \(\frac{n !}{(n-r) !}\), we have :
ⁿP_n= \(\frac{n !}{(n-n) !}\) or n! = \(\frac{n !}{0 !}\)
or 0! = \(\frac{n !}{n !}\) = 1
∴ 0! = 1.

(iv) Permutations of n objects all at a time, when n1 are alike, n2 are alike, and so on but different from n1 and n2 are given by
P = \(\frac{n !}{n_{1} ! n_{2} ! n_{3} !}\)

(v) Circular permutations : If we consider arrangements of objects in a circle, instead of a line, then we speak of circular permutations.

If n distinct objects are arranged in a circle, each object can shift position n times and returns to its original without disturbing the arrangement. Thus, each circular permutation gives n linear permutations and the number of distinguishable circular permutations, p is given by p = \(\frac{n !}{n}\) = (n – 1)!.

Combinations :
Important Definition: (i) A combination is a selection of some or all of a number of different objects. In a combination, the order of selection of the objects is immaterial.

(ii) The difference between a permutation and a combination of the objects is the “Order” does matter in a permutation, while it does not matter in case of a combination.

(iii) Relation between permutations and combinations is
C(n, r) = \(\frac{\mathrm{P}(n, r)}{r !}\), where C(n,r)
is the number of combinations of n objects taken r at a time.

or P(n,r) = (r!)C(n, r)
Note : ⁿP_r and ⁿC_r may also be denoted as P(n, r) and C(n, r) respectively.

Important Formulae:
(i) C(n, r) = \(\)
Note : (a) C(n, n)= 1 (b) C(n, 0) = \(\frac{n !}{0 !(n-0) !}\) = 1.
(ii) Complementary combinations
C (n, r) = C(n, n-r)
(;iii) Pascal’s rule
C(n, r) + C(n, r – 1) = C(n + 1, r).