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 p**otential applications**.

**Defining F****actorial but Addition**

**actorial 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 f**actorial 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:

$=8×6×4×2$

7!!$=7×5×3×1$

**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**

$5!=5×4×3×2×1=120$

5! + 5! = 120 + 120 = 240

### Example 3

**Evaluate **2! + 3! + 4!

**Solution**

$2!=2×1=2$

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**

$5!=5×4×3×2×1=120$

$6!=6×5!=720$

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**

$2!=2×1=2$

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**

$3!=3×2×1=6$

$4!=4×3×2×1=24$

$5!=5×4×3×2×1=120$

$6!=6×5×4×3×2×1=720$

$4!+5!+6!−3!=24+120+720−6=858$

**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 $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.