Permutations and Combinations - Some useful Formulae

·  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.

·  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)

·  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!.

·  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!)

·  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.

·  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:

       i.            ⁿC_n = 1 and ⁿC₀ = 1.

     ii.            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