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Permutation and Combination 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:

1. We define 0! = 1.
2. 4! = (4 x 3 x 2 x 1) = 24.
3. 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:

1. All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).
2. 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) … (nr + 1) =

Examples:

1. 6P2 = (6 x 5) = 30.
2. 7P3 = (7 x 6 x 5) = 210.
3. 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 =

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:

1. 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.
2. All the combinations formed by a, b, c taking ab, bc, ca.
3. The only combination that can be formed of three letters a, b, c taken all at a time is abc.
4. Various groups of 2 out of four persons A, B, C, D are:

AB, AC, AD, BC, BD, CD.

5. 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:

Note:

1. nCn = 1 and nC0 = 1.
2. nCr = nC(n – r)

Examples:

i.   11C4 =

ii.

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