 # Factorial

### Introduction to Factorial

One of the most standard theories of permutations and combinations is the application of factorial notation. Using the theory of factorials, many complex equations and activities can be made simpler. In math, Factorial is the product of all positive integers less than or equal to (≤) an assigned positive integer. A factorial is denoted by that integer and an exclamation point. Thus, factorial 5 is expressed as 5! , meaning that 1 × 2 × 3 × 4 × 5.  Factorial zero is described as equal to (=) 1. Even so, factorial may not seem to be much complicated, but the use of factorial notation for fractions & non-negative integers is a slightly difficult.

That being said let's Simplify Factorial.

### Simplifying Factorial Expressions

Problem solving is perhaps the best way to see mathematics in action. So here we are to take you through the problems that will significantly help you learn how to simplify factorial expressions.

In order to simplify factorial expressions using variables that're there in the numerator and denominator— we would require to produce common factors in both locations such that they can be canceled. Because, this is of course our objective.

The key is to establish a comparison between the factorials as well identify which amongst them is higher in value. Let’s say we seek to compare the factorials ({n + 4}!(n+4)! And {n + 2}!(n+ 2)!

It is pretty obvious to observe that {n + 4}! > {n + 2}! Is true for all values of ‘n’ inasmuch as the factorial is interpreted, that is to say, the items in the interior of the parenthesis is a whole number ≥ zero (0).

Thus, this suggests that we can stretch {n + 4}! Until the expression {n+2}! Takes shape into the sequence.

Like this below:-

(N+4)! = (N+3). (N+3) (N+2)!

### Simplifying Negative Integer

Let’s take for example {n - 5}! And {n - 2}! In such a situation, we are subtracting the variable by some number. Keep in mind that here the bigger expression is the one with smaller subtrahend, or the value that’s subtracted from the minuend. Hence, this suggests that {n - 2}! > {n - 5}!

This further concludes that the expression {n - 2}! Will have {n - 5}! In its expanded form

(N -2)! = (N-2). (N-3) . (N-4) . (N-5)!

### Solved Example

Problem 1

How many different ways can 7 students in a relay race come 1st, 2nd and 3rd?

Solution 1

Seeing that it is a long-drawn list, thus, if 7 students are named T, U, V, W, X, Y, Z

Then the list involves:

TUV, TUW, TUX, TUY, TUZ, TVU, TVW, TVX, TVY... etc

Using the formula – 7! (7−3)! = 7! 4!

Next, we will write the multiplications out in full:

7* 6 * 5 * 4 * 3 * 14 * 3 * 2 * 1   = 7 * 6 * 5

That was neat. The multiplication of {4 × 3 × 2 × } "invalidated", giving up only 7 × 6 × 5. And:

7 * 6 * 5 = 210

Thus, as an answer, we get 210 different ways by which 7 students secure positions at 1st, 2nd and 3rd.

And you’re done!

Problem 2

How many ways can we arrange 4 alphabets from the start (without repeating)?

Solution 2

In order to determine the equals, you need to simply multiply the positive integers altogether that are less than or equal to (≤) 4. Now, likewise

For 1 Alphabet "a" there is solely 1 way: 1×1 = (a)

For 2 alphabet "ab" there are 2 ways: 1×2= (ab, ba)

For 3 Alphabet "abc" there are 6 ways: 1×2×3 = (abc, acb, bac, bca, cab, cba)

Do you want to try for 5 Alphabets "abcde”. Go On!

### Fun Facts/ Key Takeaways

• The use of exclamation (!) was first started by a famous French Mathematician - Christian Kramp in 1808.
• factorial actually originated from combinatorics and that's one and only reason you find factorials all over combinatorics.
• Factorial had been developed as a simplified manner to express the number or a structured order of a cluster of items, which, certainly, we find by using the multiplication rule of counting.
• Factorial is though a kind of informal operation of the multiplication rule.
• double-precision numbers are only factual until fifteen digits whereas the single-precision are only accurate until eight digits.
• Bigger values of ‘n’ yield an outcome that has an appropriate alignment magnitude and is just accurate for the 1st fifteen digits.
• Bigger values of ‘n’ yield an outcome that has an appropriate alignment of magnitude and is accurate for the 1st 8 digits.

1. What is the Logic Behind Zero Factorial Equals 1?

This is the point where it gets problematic. Since we only imagine factorials in the frame of reference in which they’re generally interpreted, i.e. the “product of all the positive integers (≤)the number” then puzzling out zero (0)! is like ramming into a brick wall.

You must have heard a lot from your fellow students or may be some good at mathematics that 0! Is defined as 1, and if you ask why they would reply “because it’s defined as one”.

2. So, do we Actually Decide that Zero Factorial Equals One?

The trick is simply formula “n”!

Thus... simplifying how many ways can we arrange no alphabets? Only one way, a vacant space: arrange alphabets none

So, 0! = 1.

3. How are Factorial Notations Useful?

Factorials are frequently used in following :

• ü  assessment of permutations and combinations
• ü  advanced statistics or combinatorics
• ü  in the coefficients of terms of binomial expansions
• ü  Including non-integral values like gamma function.
• ü  Solving simple algebra to calculus
• ü  finding probabilities

4. Is it Possible to have Factorials of Decimals?

Yes we can have factorials for numbers like 0.5 or − 6.17. But for that you will require exploring into a subject matter called the "Gamma Function". And remember they can also be negative (other than integers).