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