Permutation and combination Important formulas:

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Permutation and combination Important formulas:
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:
We define 0! = 1.
4! = (4 x 3 x 2 x 1) = 24.
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. 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!.

4. 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!)

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.
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!/(r!)(n – r)! = n(n – 1)(n – 2) … to r factors / r!.

Note:
i. nCn = 1 and nC0 = 1.
ii. nCr = nC(n – r)

Examples:
i. 11C4 = (11 x 10 x 9 x 8) /(4 x 3 x 2 x 1) = 330.

ii. 16C13 = 16C(16 – 13) = 16C3
= (16 x 15 x 14) /3! = (16 x 15 x 14) /(3 x 2 x 1) = 560

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