- 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:
- 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, cby taking two at a time are (ab, ba, ac, ca, bc, cb).
- All permutations made with the letters a, b, ctaking 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:
- 6P2= (6 x 5) = 30.
- 7P3= (7 x 6 x 5) = 210.
- number of all permutations of nthings, 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, ctaking ab, bc, ca.
- The only combination that can be formed of three letters a, b, ctaken 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 abba 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:
- . nCn= 1 and nC0 = 1.
- nCr= 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 |