PERMUTATIONS AND COMBINATIONS
Frequently Asked Questions
A permutation is an arrangement of objects in a definite order. If the order of selection changes, the permutation changes.
A combination is a selection of objects where order is not important. Different orders of the same objects represent the same combination.
In permutation, order matters; in combination, order does not matter.
The number of permutations is \(^{n}P_{r} = \dfrac{n!}{(n-r)!}\).
The number of combinations is \(^{n}C_{r} = \dfrac{n!}{r!(n-r)!}\).
The factorial of \(n\), written as \(n!\), means the product \(n \times (n-1) \times (n-2) \times \cdots \times 1\).
By definition, \(0! = 1\).
This definition ensures the validity of formulas such as \(^{n}P_{n} = n!\) and \(^{n}C_{0} = 1\).
The value of \(^{n}P_{n}\) is \(n!\), which represents all possible arrangements of \(n\) objects.
Both \(^{n}C_{0}\) and \(^{n}C_{n}\) are equal to 1.
For all integers \(n\) and \(r\), \(^{n}C_{r} = {}^{n}C_{n-r}\).
They are related by \(^{n}P_{r} = {}^{n}C_{r} \times r!\).
A linear permutation is an arrangement of objects in a straight line.
A circular permutation is an arrangement of objects around a circle, where relative positions matter.
The number of circular permutations is \((n-1)!\).