Fundamental Principle of Counting

If an event can occur in m different ways, following which another event can occur in n different ways, following which a third event can occur in p different ways, then the total number of occurrence to the events in the given order is m × n × p.

Factorial

The Product of first n natural numbers is known as Factorial. It is denoted by n!.

n! = n.(n-1).(n-2)...3.2.1

n! = n.(n-1)! = n(n-1)(n-2)!

0! = 1

Permutation (Arrangement)

A Permutation is an arrangement in a definite order of a number of distinct objects taking some or all at a time.

Case 1

The number of permutations of n different objects taken r at a time, and the objects do not repeat is n(n–1)(n–2)...(n–r+1), which is denoted by nPr.

nPr = n!/(n-r)!, 0 ≤ r ≤ n

nPn = n!

Case 2

The number of permutations of n different objects taken r at a time, where repetition is allowed, is nr.

Case 3

The number of permutations of n objects, where p objects are of the same kind and rest are all different = n!/p!

Combination (Selection)

The number of combinations of n different things taken r at a time, denoted by nCr, is given by

nCr = n!/r!(n-r)!, 0 ≤ r ≤ n

nPr = nCr r!

nCr + nCr-1 = n+1Cr