Permutation & combination
Definition of Factorial
Factorial of a number n can be defined as product of all positive numbers less than or equal to n. It the multiplying sequence
of numbers in a descending order till 1. It is defined by the symbol of exclamation (!).
1!=1
2!=2×1
3!=3×2×1
4!=4×3×2×1
5!=5x4x3x2x1
...
n! = n×(n-1)×(n-2)...×2×1
Remember 0! = 1.
Definition of Permutation: Permutation is defined as the arrangement
of r things that can be done out of total n things.
Permutation basically used for arrangement , arrangement can be horizontal
, vertical or circular.
Formula of Permuation
Permutation is denoted by nPr it is equal to n!/(n-r)!
Defination of Combination: Combination is defined as selection of r things that can be done out of total
n things.
Combination can be used in the case of selection of team or committee or group, besically it is used for selection
procedure .
Formula of Combination
Combination is denoted nCr
it is equal to n!/r!(n-r)!
Fundamental principle of counting
Product Rule:
If an activity A can be done in “m” ways and an another activity “B” can be done in “n” ways,
A and B together can be done in “m x n” ways
Addition Rule:
If an activity A can be done in “m” ways and an another activity “B” can be done in “n” ways,
then A or B together can be done in “m + n” ways
Important results on Permutations
(1) np n-1 = np n
(2) np n = n!
(3) np r = nn-1p r-1
(4) np 0 = 1
(5) np 1 = n
(6) When in a permutation of n things taken r at a time, a particular thing always occurs.
Required number of permutation will be = (r)n-1p r-1
(7) The number of permutations of n different things
taken r at a time, when a particular object is never taken in any arrangement is n-1p r
(8) The number of things taken all at a time, in which p1 things are alike of 1st kind, p2 things are alike of 2nd kind,
pr things are alike of rth kind and so on.
Required number of permutation will be =
n!/(p1!p2!-------pr!)
(9) . Circular Permutation
Circular Permutations are the permutations of things along the circumference of a circle . It is important to note that is a circular arrangement
, there is neither a beginning (first term) nor an end (the last term) .
the number of Circular Permutation depends on the relative position
of the objects, we fix the position of one object and then arrange the rest (n-1) objects in all possible ways .
The number of circular
permutations of n different object is (n-1)!
Important Result on Combination
(1) n C n = 1
(2) n C 0 = 1
(3) n Pr =
r! x n C r
(4) n C x = n C y, then either x = y or
x + y = n
(5) n C r + n C r + 1 = n + 1 C r + 1
(6) The total number of combinations of n dissimilar things, takingany number of them at a time, when all the things are
different .
n C 0 + n C 1 + n C 2 + -----------+
n C n = 2n
n C 1 + n C 2 + n C 3 + -----------+
n C n = 2n - 1
