Let n be a positive integer. Then, factorial n is denoted by n! is defined as:
n! = n(n - 1)(n - 2)(n - 3).....3.2.2.
(i) 6! = (6 x 5 x 4 x 3 x 2 x 1) = 720
(ii) 5! = (5 x 4 x 3 x 2 x 1) = 120
(ii) 4! = (4 x 3 x 2 x 1) = 24
we define, 0! = 1.
The different arrangements of a given number of things by taking some or all at a time, are called permutations.
All permutations made by with 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 objects is: nPr = n(n - 1)(n - 2)..... (n- r -1) =
(n - r)!
(i) 9P3 = (9 x 3) = 27.
(ii) 8P5 = (8 x 5) = 40. Note: 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 pr are alike of rth kind,
such that (p1 + 2 + ..... r) = n.
Then, number of permutations of these n objects is =
(p1!)(p2!) ... (pr!)
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.
(i) Suppose there are three boys A,B, C, and we want to select two out of three, then the combinations are: AB, BC, and CA.
Note: AB and BA represent the same selection.
(ii) All the combinations formed by a, b, c, taking two at a time are ab, bc, ca.
(iii) The only combination that can be formed of three letters a, b, c taken all at a time is abc.
(iv) Various groups of 2 out of four persons A, B, C, D are:
AB, AC, AD, BC, BD, CD.
(v)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:
r!(n - r)!
n(n - 1)(n - 2).....to r factors
Note: nCn = 1 and nC0 = 1. An Important Result: nCr = nC(n - r)