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Quantitative Aptitude :: Permutation and Combination

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.

6! = (6 x 5 x 4 x 3 x 2 x 1) = 720.

7! = (7 x 6 x 5 x 4 x 3 x 2 x 1) = 5040.



2. Permutations:


The different arrangements of a given number of things by taking some or all at a time, are called permutations.

Examples:



\( \mathbb{i.}\) All permutations (or arrangements) made with the letters a, b, c by taking two at a time are (ab, ba, ac, ca, bc, cb).

\( \mathbb{ii.}\) 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:\( ^{n}P_{r} \) = n(n - 1)(n - 2) ... (n - r + 1) = \(\frac{n!}{(n - r)!}\)


Examples:

\( \mathbb{i.}\)\( ^{6}P_{2} \) = (6 x 5) = 30.

\( \mathbb{ii.}\)\( ^{7}P_{3} \) = (7 x 6 x 5) = 210.

\( \mathbb{iii.}\)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 = \(\frac{n!}{(p_{1}!).(p_{2})!.....(p_{r}!)}\)

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:


\( \mathbb{i.}\) 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.

\( \mathbb{ii.}\) All the combinations formed by a, b, c taking ab, bc, ca.

\( \mathbb{iii.}\) The only combination that can be formed of three letters a, b, c taken all at a time is abc.

\( \mathbb{iv.}\) Various groups of 2 out of four persons A, B, C, D are: AB, AC, AD, BC, BD, CD.

\( \mathbb{v.}\) 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:\(^{n}C_{r} = \frac{n!}{(r!)(n - r)!} = \frac{n(n - 1)(n - 2) ... to r factors}{r!}\)



Important Note:

\( \mathbb{i.}\) \(^{n}C_{n} = 1\) and \(^{n}C_{0} = 1\).
\( \mathbb{ii.}\) \(^{n}C_{r} = ^{n}C_{n-r}\)