Question

What is the permutation rule?

Answer 1

Hint: A permutation is one of all possible ordered arrangements of the elements. The set of elements may conform to certain rules to aid our discussion.

Complete asnwer:
Let’s examine the permutation of three-digit numbers derived from the set of ten natural numbers 0-9.
We know from our prior knowledge of the base 10 system that there are a thousand distinct values that can be represented. Therefore, there are 10 possibilities for the units place, 10 possibilities for the tens place and 9 possibilities for the hundreds place (excluding the digit 0 as we are required to calculate the 3 digit number). Hence, the total number of words comes out to be \[9 \times 10 \times 10 = 900\]
Therefore, it can be said that the notion “permutation” relates to the act of arranging all the members of a set into some sequence or if the set is already ordered, rearranging its elements, a process called permuting.
Mathematically, it can be written as
 $\Rightarrow P(n,r) = \dfrac{{n!}}{{(n - r)!}} $
P is the number of permutations.
n is the total number of objects in the set.
r is the number of choosing objects from the set.

Note: Similar to the concept of Permutation, there is a concept of Combination which is the selection of items in which order does not matter. Number of ways of selection of r items (order does not matter) out of n things at a time is given by
 $ C(n,r) = \dfrac{{n!}}{{(n - r)!r!}} $
Where, C is the number of combinations.
n is the total number of items in the set.
r is the number of choosing objects in the set.
Also,
Number of different permutations of n objects where there are $ {n_1} $ repeated items, $ {n_2} $ repeated items,...... $ {n_k} $ repeated items is given by
 $ P = \dfrac{{n!}}{{{n_1}!{n_2}!......{n_k}!}} $