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.
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
A Permutation is an arrangement in a definite order of a number of distinct objects taking some or all at a time.
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!
The number of permutations of n different objects taken r at a time, where repetition is allowed, is nr.
The number of permutations of n objects, where p objects are of the same kind and rest are all different = n!/p!
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