OverviewExercise"Nothing is impossible to a willing heart."
 (Proverb)


Factorial Notation:
Let n be a positive integer. Then, factorial n, denoted n! is defined as:
n! = n(n  1)(n  2) ... 3.2.1.
Examples:
We define 0! = 1.
4! = (4 x 3 x 2 x 1) = 24.
5! = (5 x 4 x 3 x 2 x 1) = 120.

Permutations:
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
Examples:
All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
All permutations made with the letters a, b, c taking all at a time are:
( abc, acb, bac, bca, cab, cba)

Number of Permutations:
Number of all permutations of n things, taken r at a time, is given by:
^{n}P_{r} = n(n  1)(n  2) ... (n  r + 1) = 
n! 
(n  r)! 
Examples:
^{6}P_{2} = (6 x 5) = 30.
^{7}P_{3} = (7 x 6 x 5) = 210.
Cor. number of all permutations of n things, taken all at a time = n!.

An Important Result:
If there are n subjects of which p_{1} are alike of one kind; p_{2} are alike of another kind; p_{3} are alike of third kind and so on and p_{r} are alike of r^{th} kind, such that (p_{1} + p_{2} + ... p_{r}) = n.
Then, number of permutations of these n objects is = 
n! 
(p_{1}!).(p_{2})!.....(p_{r}!) 

Combinations:
Each of the different groups or selections which can be formed by taking some or all of a number of objects is called a combination.
Examples:

Suppose we want to select two out of three boys A, B, C. Then, possible selections are AB, BC and CA.
Note: AB and BA represent the same selection.
All the combinations formed by a, b, c taking ab, bc, ca.
The only combination that can be formed of three letters a, b, c taken all at a time is abc.

Various groups of 2 out of four persons A, B, C, D are:
AB, AC, AD, BC, BD, CD.
Note that ab ba are two different permutations but they represent the same combination.

Number of Combinations:
The number of all combinations of n things, taken r at a time is:
^{n}C_{r} = 
n! 
= 
n(n  1)(n  2) ... to r factors 
. 
(r!)(n  r)! 
r! 
Note:
^{n}C_{n} = 1 and ^{n}C_{0} = 1.
^{n}C_{r} = ^{n}C_{(n  r)}
Examples:
i. ^{11}C_{4} = 
(11 x 10 x 9 x 8) 
= 330. 
(4 x 3 x 2 x 1) 
ii. ^{16}C_{13} = ^{16}C_{(16  13)} = ^{16}C_{3} = 
16 x 15 x 14 
= 
16 x 15 x 14 
= 560. 
3! 
3 x 2 x 1 

