- 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.
- 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:
nPr = n(n – 1)(n – 2) … (n – r + 1) = n! (n – r)!
- 6P2 = (6 x 5) = 30.
- 7P3 = (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 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!)
- 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:
nCr = n! = n(n – 1)(n – 2) … to r factors . (r!)(n – r)! r!
- nCn = 1 and nC0 = 1.
- nCr = nC(n – r)
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