Question

Build a generating function for ar, the number of distributions of r identical objects into (a) Five different boxes with at most three objects in each box (b) Three different boxes with between three and six objects in each box (c) Six different boxes with at least one object in each box (d) Three different boxes with at most five objects in the first box

Answer 1

a) ar = ( 1 + x + x^2 + x^3)^5

b) ar = ( x^3 + x^4 + x^5 + x^6 )^3

c) ar = ( x^1 + x^2 + x^3 + x^4 + ....)^6

d) ar = ( 5 + x^1 + x^2 + x^3 + x^4 + x^5 )^3

Explanation:

Solution:-

- The generating function (ar), where the number of (r) identical objects.

- The number of identical boxes = r

- The function parameter, the number of different boxes = n

- The number of objects in each box = k

- The general generating function (ar) is of the form:

ar = (x^0 + x^1 + x^2 + x^3 + x^4 + ....+ x^k)^n

part a)

- We have 5 different boxes, n = 5.

- We are to place at most 3 objects in each box, k ≤ 3

- The generating function would be:

ar = ( 1 + x + x^2 + x^3)^5

part b)

- We have 3 different boxes, n = 3.

- We are to place 3 to 6 objects in each box, (3 ≤ k ≤ 6)

- The generating function would be:

ar = ( x^3 + x^4 + x^5 + x^6 )^3

part c)

- We have 6 different boxes, n = 6.

- We are place at-least 1 objects in each box, k ≥ 1

- The generating function would be:

ar = ( x^1 + x^2 + x^3 + x^4 + ....)^6

part d)

- We have 3 different boxes, n = 3.

- We are place at-most 5 objects in each box, k ≤ 5

- The generating function would be:

ar = ( 5 + x^1 + x^2 + x^3 + x^4 + x^5 )^3

Answer 2

a) (1 + x + x^2 + x^3) ^5

b) (x^3 + x^4 + x^5 +x^6) ^3

c) ( x + x^2 + x^3 + x^4..........) ^6

d) ( 1 + x + x^2 + x^3 + x^4 + x^5) ^3

Explanation:

A generating function is a process of encoding an infinite sequence of numbers (ar) by giving them a treatment as the coefficients of a power series. This formal power series is the generating function. As opposed to an ordinary series, this formal series is allowed to diverge, implying that the generating function is not always a true function and the "variable" is typically an indeterminate.

From the information above, build a generating function for ar, the number of distribution of r identical objects into:

(a) 5 different boxes with at most three objects in each boxes, this would be done as follows:

Answer = (1 + x + x^2 + x^3) ^5

(b) Three different boxes with between three and six objects in each boxes.

The answer is:

Answer= (x^3 + x^4 + x^5 +x^6) ^3

(c) Six different boxes with at least one object in each box.

The answer is:

Answer= ( x + x^2 + x^3 + x^4..........) ^6

(d) Three different boxes with at most five objects

The answer is:

Answer = ( 1 + x + x^2 + x^3 + x^4 + x^5) ^3