Permutation ☰

In mathematics, a permutation is a rearrangement of a set of objects in a specific order. Permutations are used to count the number of ways that a set of objects can be arranged. A permutation of a set S is a one-to-one and onto mapping of S onto itself.
More formally, a permutation of a set S is a bijective function \( \sigma \): \(S \rightarrow S \) . In other words, \(\sigma \) (Sigma) is a function that maps every element in S to a unique element in S, and every element in S is mapped to exactly once. We can represent a permutation \( \sigma \) of a set S by writing down its values in a particular order, for example:
\( \sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 1 & 4 & 2 \end{pmatrix} \)

This notation means that \( \sigma(1) = 3 \), \( \sigma(2)=1 \), \(\sigma(3)=4 \) and \( \sigma(4)=2 \). In other words, the first row represents the elements of S in their original order, and the second row represents their order after applying the permutation \( \sigma \).
The number of permutations of a set S with \(n\) elements is denoted by \(n!\), which is read as "n factorial". The factorial function is defined as:
\( n!=n \cdot (n-1) \cdot (n-2) \ldots 2 \cdot 1 \)

For example, \(5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120 \) , which means that there are 120 permutations of a set with 5 elements.

Permutations can be used to solve various counting problems. For example, suppose we have 5 different books and we want to arrange them on a shelf. The number of ways to arrange the books is given by the number of permutations of a set with 5 elements, which is \( 5!=120 \).

Another example is the number of ways to select a committee of 3 people from a group of 10 people. The number of ways to select the committee is given by the number of permutations of a set with 10 elements taken 3 at a time, which is denoted by \(_{10} P_3 \) and is calculated as:
\( _{10} P_3 =\frac{10!}{(10-3)!} = 10 \cdot 9 \cdot 8 =720 \)

In general, the number of permutations of a set with \(n\) elements taken \(r\) at a time is denoted by \(_n P_r\) and is calculated as:
\( _n P_r= \frac{n!}{(n-r)!} \)

In conclusion, permutations are a fundamental concept in combinatorics and are used to count the number of ways that a set of objects can be arranged. The number of permutations of a set with n elements is \(n!\), and the number of permutations of \(r\) elements taken from a set of \(n\) elements is \(_n P_r\).

Combination ☰

In mathematics, a combination is a way of selecting items from a larger set without regard to the order in which they are selected. It is denoted by \( _nC_k \), where \(n\) is the number of items in the set and \(k\) is the number of items being selected. A combination is also known as a binomial coefficient.
The formula for the number of combinations is given by:
\( {n \choose k} = \frac{n!}{k!(n-k)!} \) , where \(n!\) denotes the factorial of \(n\), which is the product of all positive integers from 1 to \(n\), and \(0!\) is defined to be 1. The notation \( {n \choose k} \) is read as "\(n\) choose \(k\)".

For example, suppose we have a set of five numbers \({1,2,3,4,5}\). We want to select three numbers from this set without regard to order. The number of combinations of size 3 that can be chosen from this set is:
\( \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \cdot 4 \cdot 3}{3 \cdot 2 \cdot 1} = 10 \)
Therefore, there are 10 ways to select three numbers from the set \({1,2,3,4,5} \).
Combinations are useful in a variety of mathematical and real-world situations. For example, they can be used to count the number of ways to form a committee of a certain size from a group of people, to calculate the probabilities of certain events occurring in probability theory, and to analyze the outcomes of certain games in game theory.

It is important to note that the number of combinations is always less than or equal to the number of permutations, which are the ways of selecting items from a set while considering their order. The formula for permutations is given by:
\(P(n,k) = \frac{n!}{(n-k)!} \) , where \(P(n,k)\) denotes the number of permutations of size \(k\) that can be chosen from a set of size \(n\).

In summary, a combination is a way of selecting items from a larger set without regard to order, and the number of combinations is given by the formula \( {n \choose k} = \frac{n!}{k!(n-k)!} \). Combinations are useful in a variety of mathematical and real-world situations, and they are related to permutations, which are the ways of selecting items from a set while considering their order.