Question

HowProve Theorem 4 by first setting up a one-to-one correspondence between permutations of\({\rm{n}}\)objects with\({{\rm{n}}_{\rm{i}}}\)indistinguishable objects of type\({\rm{i,i = 1,2,3,}}....{\rm{,k}}\), and the distributions of\({\rm{n}}\)objects in k boxes such that\({{\rm{n}}_{\rm{i}}}\)objects are placed in box\({\rm{i,i = 1,2,3,}}....{\rm{,k}}\)and then applying Theorem 3.

Short answer

\({{\rm{S}}_{\rm{1}}}\)Represent the set that contains those \({{\rm{n}}_{\rm{1}}}\)distinguishable objects for\({\rm{i = 1,2, \ldots ,k}}\).

Step-by-step solution

Concept Introduction

Counting is the act of determining the quantity or total number of objects in a set or a group in mathematics. To put it another way, to count is to say numbers in sequence while giving a value to an item in a group on a one-to-one basis. Objects are counted using counting numbers.

 Prove Theorem 4

Let \({{\rm{S}}_{\rm{1}}}\)represent the set that contains those \({{\rm{n}}_{\rm{1}}}\) distinguishable objects for \({\rm{i = 1,2, \ldots ,k}}\)in any distribution of $n$ distinguishable objects into \({\rm{k}}\)boxes where the \({\rm{i}}\)-th box gets \({{\rm{n}}_{\rm{1}}}\)number of objects. Now the collection of items may be characterised as\({\rm{\`E }}_{{\rm{i = 1}}}^{\rm{k}}{{\rm{S}}_{\rm{i}}}\), with \(\left| {{{\rm{S}}_{\rm{1}}}} \right|{\rm{ = }}{{\rm{n}}_{\rm{1}}}\)the number of objects of type \({\rm{1,}}\left| {{{\rm{S}}_{\rm{2}}}} \right|{\rm{ = }}{{\rm{n}}_{\rm{2}}}\)the number of objects of type 2, and so on. Type \({\rm{i}}\)denotes things that go into the \({\rm{i}}\)-th box and are hence indistinguishable from one another. By simply permuting these sets\({{\rm{S}}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,k}}\), all potential combinations of object distribution may be found, and the number of ways to distribute the objects is the same as

The number of such permutations, which is

\(\frac{{{\rm{n!}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{!}}{{\rm{n}}_{\rm{2}}}{\rm{!L}}{{\rm{n}}_{\rm{k}}}{\rm{!}}}}\)

Let \({{\rm{S}}_{\rm{1}}}\)represent the set that contains those \({{\rm{n}}_{\rm{1}}}\)distinguishable objects for \({\rm{i = 1,2, \ldots ,k}}\)in any distribution of \({\rm{n}}\)distinguishable objects into \({\rm{k}}\) boxes where the \({\rm{i}}\)-th box gets \({{\rm{n}}_{\rm{i}}}\)number of objects.