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1.  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:
 
i.   We define 0! = 1.  
ii.   4! = (4 x 3 x 2 x 1) = 24.  
iii.   5! = (5 x 4 x 3 x 2 x 1) = 120.  
2.  Permutations:
  The different arrangements of a given number of things by taking some or all at a time, are called permutations.
  Examples:
 
i.   All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).  
ii.   All permutations made with the letters a, b, c taking all at a time are:
( abc, acb, bac, bca, cab, cba)		  
3.  Number of Permutations:
  Number of all permutations of n things, taken r at a time, is given by:
 
  nPr = n(n - 1)(n - 2) ... (n - r + 1) =
n!
(n - r)!
 
  Examples:
 
i.   6P2 = (6 x 5) = 30.  
ii.   7P3 = (7 x 6 x 5) = 210.  
iii.   Cor. number of all permutations of n things, taken all at a time = n!.  
4.  An Important Result:
  If there are n subjects of which p1 are alike of one kind; p2 are alike of another kind; p3 are alike of third kind and so on and pr are alike of rth kind,
such that (p1 + p2 + ... pr) = n.
 
Then, number of permutations of these n objects is =
n!
(p1!).(p2)!.....(pr!)
   
5.  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:
 
1.    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.  
2.    All the combinations formed by a, b, c taking ab, bc, ca.  
3.    The only combination that can be formed of three letters a, b, c taken all at
a time is abc.		  
4.    Various groups of 2 out of four persons A, B, C, D are:  
   
  AB, AC, AD, BC, BD, CD.    
 
5.    Note that ab ba are two different permutations but they represent the same combination.  
6.  Number of Combinations:
  The number of all combinations of n things, taken r at a time is:
 
  nCr =
n!
(r!)(n - r!)
=
n(n - 1)(n - 2) ... to r factors
r!
   
  Note:
 
i.   nCn = 1 and nC0 = 1.  
ii.   nCr = nC(n - r)  
  Examples:
 
i.  
11C4 =
(11 x 10 x 9 x 8)
(4 x 3 x 2 x 1)
 = 330  
 
ii.  
16C13 = 16C(16 - 13) = 16C13 =
16 x 15 x 14
3!
=
16 x 15 x 14
3 x 2 x 1
 = 560.
 
 
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