Permutation and Combination - Important Formulas |Aptitude

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:
   1. We define 0! = 1.
   2. 4! = (4 x 3 x 2 x 1) = 24.
   3. 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:
   1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
   2. 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:
   1. ⁶P₂ = (6 x 5) = 30.
   2. ⁷P₃ = (7 x 6 x 5) = 210.
   3. 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 p_r are alike of r^th kind, 
   such that (p₁ + p₂ + ... p_r) = 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!	=	n(n - 1)(n - 2) ... to r factors	.
   	(r!)(n - r)!		r!

   Note:
   1. ⁿC_n = 1 and ⁿC₀ = 1.
   2. ⁿC_r = ⁿC_{(n - r)}
   Examples:

   i.   11C4 =	(11 x 10 x 9 x 8)	= 330.
   	(4 x 3 x 2 x 1)

   ii.   16C13 = 16C(16 - 13) = 16C3 =	16 x 15 x 14	=	16 x 15 x 14	= 560.
   	3!		3 x 2 x 1