Factorial but Addition – Definition, Applications, and Examples

Welcome to the world of “factorial but addition.” or “factorial by addition.” This concept may not be as mainstream as its multiplicative counterpart, but it opens up an intriguing and lesser-explored dimension of mathematical analysis. In this article, we dive deep into this notion, unraveling its definition, properties, and potential applications.

Defining Factorial but Addition

The “factorial but addition” is defined as the sum of all integers from 1 to n. This is more commonly known as the arithmetic series of the first n natural numbers. Its formula is given by:

$S_n$​​=n (n+1) / 2​

Where $s_n$​​​ represents the sum of the first $n$ natural numbers.

The standard factorial, denoted by “n!“, is the product of all positive integers less than or equal to n. When we talk about “factorial by addition,” it seems we’re moving from a multiplicative view of factorials to an additive one.

For example, the “factorial by addition” for 5 would be:

$S_5$​​=5(5+1)/2 = 30/2 =15

This definition provides an alternate perspective on factorial, focusing on cumulative addition rather than cumulative multiplication. However, it’s essential to understand that this concept doesn’t replace the traditional factorial; it offers a different angle of looking at sequence summation.

Properties

Value of 0!

As mentioned earlier, by definition, 0! is equal to 1. This might seem counterintuitive, but it simplifies many mathematical formulas and is consistent with the definition of the factorial function.

Growth Rate

The factorial function grows very fast. Even for relatively small numbers, their factorials can be huge. This exponential growth is due to the multiplicative nature of factorials.

Relation with Permutations

Factorials describe the number of ways to arrange n distinct items. For example, there are n! ways to arrange n distinct books on a shelf.

Recursive Definition

n! = n * (n-1)!.

This property is often used in algorithms, especially when writing recursive programs to compute factorials.

Division of Factorials

For any positive integers $m$ and $n$ where m > n:

m! / n! = m * (m-1) * … * (n+1).

Product of Factorials

Given two positive integers $m$ and $n$:

m! * n!

This isn’t simplifiable to a single factorial, but it’s noteworthy to understand that the product of two factorials remains a product of integers in sequential order, albeit with repetition.

Double Factorial

The double factorial, denoted as n!!, is the product of integers of the same parity as n less than or equal to n. For instance:

$=8\times 6\times 4\times 2$

7!!$=7\times 5\times 3\times 1$

Binomial Coefficient

Factorials are used in the computation of combinations or binomial coefficients. The number of ways to choose k items from n items without repetition and without order is given by:

n choose k = n! / (k! * (n-k)!)​

Exercise 

Example 1

Evaluate 3! + 4!

Solution

3! = 3 × 2 × 1 = 6

4! = 4 × 3 × 2 × 1 = 24

3! + 4! = 6 + 24 = 30

Example 2

Evaluate 5! + 5!

Solution

$5!=5\times 4\times 3\times 2\times 1=120$

5! + 5! = 120 + 120 = 240

Example 3

Evaluate 2! + 3! + 4!

Solution

$2!=2\times 1=2$

3! = 3 × 2 × 1 = 6

4! = 4 × 3 × 2 × 1 = 24

2! + 3! + 4! = 2 + 6 + 24 = 32

Example 4

Evaluate$0!+1!+2!$

Solution

$0!=1$

1! = 1

2! = 2 × 1 = 2

0! + 1! + 2! = 1 + 1 + 2 = 4

Example 5

Evaluate 6! + 7! − 5!

Solution

$5!=5\times 4\times 3\times 2\times 1=120$

$6!=6\times 5!=720$

7! = 7 × 6! = 5040

6! + 7! − 5! = 720 + 5040 − 120 = 5640

Example 6

Evaluate 3! + 3! + 3! + 3!

Solution

3! = 3 × 2 × 1 = 6

3! + 3! + 3! + 3! = 6 + 6 + 6 + 6 = 24

Example 7

Evaluate 2! + 4! + 6!

Solution

$2!=2\times 1=2$

4! = 4 × 3 × 2 × 1 = 24

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

2! + 4! + 6! = 2 + 24 + 720 = 746

Example 8

Evaluate 4! + 5! + 6! − 3!

Solution

$3!=3\times 2\times 1=6$

$4!=4\times 3\times 2\times 1=24$

$5!=5\times 4\times 3\times 2\times 1=120$

$6!=6\times 5\times 4\times 3\times 2\times 1=720$

$4!+5!+6!-3!=24+120+720-6=858$

Applications

• Mathematics

  • Series Involving Factorials: The exponential function can be expanded using a series with factorials in denominators.
  • Binomial Coefficients: Factorials help determine coefficients in the binomial expansion. The binomial coefficient is given by (kn​)=k! (n−k) !n!​.
  • Number Theory: In problems concerning the sum of divisors or permutations of divisors, factorials might appear.

• Computer Science

  • Algorithms: Algorithms related to permutations and combinations rely on factorials to understand the number of possible outcomes.
  • Recursive Functions: The factorial function serves as an introductory example of recursion due to its self-referential nature.
  • Dynamic Programming: For solving problems in smaller steps, previously computed factorial values might be reused and summed.

• Physics and Engineering

  • Statistical Mechanics: In studying the behavior of particles in thermodynamic systems, factorials can represent the number of possible microstates.
  • Quantum Mechanics: Factorials occasionally arise in quantum computations and probability determinations.
  • Signal Processing: In analyzing or designing certain signals, factorials can be relevant in coefficient calculations.

• Biology

  • Genetics: When predicting genetic trait probabilities, factorials help in counting the combinations.
  • Population Dynamics: In models predicting breeding scenarios or certain growth patterns, factorials can be involved.
  • Species Sampling: In biodiversity studies, researchers might use factorials to determine the combinations in which species appear in samples.

• Economics and Social Sciences

  • Decision Trees: When analyzing decisions that have sequences or multiple outcomes, factorials can represent ordering.
  • Game Theory: For determining strategies in sequential games, factorials might be used.
  • Actuarial Sciences: Factorials, especially combined with addition, help in understanding insurance risks, determining life tables, or modeling various sequences of events in the financial sector.
  • Operations Research: Problems like task scheduling, optimization, and routing might require factorials to understand orderings.
  • Finance: In portfolio management, factorials can indicate possible asset combinations, especially when determining risk.

• Chemistry

  • Stereoisomerism: For molecules with multiple chiral centers, factorials are used to determine the number of possible stereoisomers.

• Medical Research

  • Combination Drug Therapies: In studying drug efficacy, especially in combination therapies, factorials can indicate the number of ways drugs can be combined.

• Cryptography

  • Permutations in Encryption: Factorials are essential in methods that rely on data permutations for encryption. Different encryption keys or methods might be summed or combined.

• Ecology

  • Biodiversity Sampling: When assessing biodiversity, factorials can assist in understanding the numerous ways species combinations can manifest in samples.

• Star Clusters

  In some advanced models, the arrangement or grouping of stars within clusters or galaxies can involve factorials to represent certain combinations or sequences.