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:
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.
2. 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)
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:
i.
6P
2 = (6 x 5) = 30.
ii.
7P
3 = (7 x 6 x 5) = 210.
iii. 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 p
1 are alike of one kind; p
2 are alike of another kind; p
3 are alike of third kind and so on and pr are alike of r
th kind,
such that (p
1 + p
2 + ... 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:
i. 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.
ii. All the combinations formed by a, b, c taking 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.
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:
i.
nC
n = 1 and
nC
0 = 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 |
