Introduction

On Unique-Factorization Combinatorics via the Pascal Triangle

Section I - Introduction

This is the 4th paper in a series of six (6) papers under the umbrella of the Title Paper :

with the 6 papers here being :

1st paper	* *	On Prime Number Systems and Core Structural Pattern Propagations
2nd paper	* *	On Counting Natural Numbers up to [ N = 1543 ]
3rd paper	* *	On Counting [ Candidates-for-Primes modulo 30 ] up to [ N =1543 ]
4th paper	* *	On Unique-Factorization Combinatorics via the Pascal Triangle
5th paper	* *	On the Match-Up of Structural Patterns within Prime Number Systems
6th paper	* *	On the Unique-Factorization Equation within each Prime Number System
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And in this paper , we shall attempt answer this question :

QUESTION :
- IF :
  - we define { Category T Composite Numbers } as natural numbers
    
    that are the product of [ T ] prime-numbers ,
  THEN :
  - how many distinct-and-different { Category T Composite Numbers } can we formulate
    
    with a set of [ H ] distinct-and-different prime-numbers ;
    - with repetition of prime-numbers as [ prime-factors ] being allowed
      - throughout the entire formulation process .
This is a necessary and preliminary step ahead of further analysis on :
- the 2nd paper : On Counting Natural Numbers up to [ N = 1543 ]
  
  and
- the 3rd paper : On Counting { Candidates-for-Primes modulo 30 } up to [ N = 1543 ] .