Factorial

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Factorial is a quantity defined for all non-negative integers. Note that zero is an integer which is neither positive nor negative.
If 'n' is an integer greater than or equal to zero, the factorial of 'n' is the product of all positive integers less than or equal to n.
It is denoted commonly by n! and occasionally by . It is read as "factorial n".
We shall use the exclamation symbol (!) hereafter.
The factorial of zero i.e., 0! is defined as 1.
Factorial of negative integers and fractions is not defined.

Example:
3! = 3 × 2 × 1 = 6
6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 36,28,800

As 'n' increases the value of n! increases very fast. See adjacent table.

Brain Teaser

n vs n!

The graph is shown only upto n = 4 which is exponential in nature. The rate of growth is much faster with increasing 'n'.
n vs ln(n!)

For higher values of 'n', natural logarithm of n! is taken on the y-axis. The graph is approximately linear (for all reasonable values of n).

Graphs of a factorial function

The increase (or rate of growth) of a factorial is faster than all polynomials and exponentials.
The graphs of n vs n! and n vs ln(n!), where 'ln' represents natural logarithm, are shown adjacent.
Since the symbol, Π is used to represent product of numbers, the factorial function can be represented as

n!	=
	=	1 . 2 . 3 . . . . (n – 2) . (n – 1) . n
	=	n . (n – 1) . (n – 2) . . . . 3 . 2 . 1
Since (n – 1)! = (n – 1) . (n – 2) . . . . 3 . 2 . 1 we have
n!	=	n . (n – 1)!
	=	n . (n – 1) . (n – 2)!
	=	n . (n – 1) . (n – 2) . (n – 3)! etc.

Example:

7!	=	7 × 6!
50!	=	50 × 49 × 48!
99!	=	99 × 98 × 97 × 96!

6 weeks = 10! seconds

Let us convert six weeks into seconds.

6 weeks	=	6 × 7 × 24 × 60 × 60 seconds
	=	6 × 7 × (2 × 3 × 4) × (5 × 2 × 2 × 3) × (10 × 2 × 3)
	=	10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
	=	10!
So 6 weeks = 10! seconds! (The symbol after 'seconds' is purely exclamatory).

Trivia

(i) There are about 60! atoms in the universe.
(ii) We know Googol = 10¹⁰⁰
We can evaluate 70! as 1.197857 × 10¹⁰⁰.
i.e, 70! > 1 Googol
So the least integer whose factorial is greater than Googol is 70.

(iii) Shuffle a deck of cards and place it on a table. The top card could be any one of the 52 cards.
The card below it could be any one of the remaining 51 cards. And so on.
In this way, you can have this number of combinations :
52 × 51 × 50 × 49 × . . . . × 3 × 2 × 1
i.e., 52! combinations.
52! ≃ 8.06587175 . . . . × 10⁶⁷
which is a very, very, very large number!
So it is extremely likely that you are the first person ever to shuffle the cards in that particular order. (So will be with any other person).

Example
Consider the word 'ASSASIN' The number of letters (inclusive of repetitions), n = 7 'A' is repeating twice i.e., m₁ = 2 'S is repeating thrice i.e., m₂ = 3 I and N are there only once i.e., m₃ = m₄ = 1 ∴ The no.of multiset permutations is given by
Physical interpretation of factorial
n! is the number of ways that 'n' unique items can be arranged. Ex : Consider the 3 letters x, y and z. They can be arranged in 6 ways as: xyz, xzy, yxz, yzx, zxy, zyx i.e., 3! ways (since 3! = 3.2.1 = 6)

Applications of factorial

Permutations and combinations in mathematics is one area where factorial is extensively used. Other areas are probability, exponentials, number theory, calculus and physics.

i. Permutations and combinations

a. Linear permutations:

Consider arrangements of a fixed length of elements (say k) taken from a set of 'n' elements.
The numbers of such k-permutations of 'n' is given by
n!/(n – k)!

b. Circular permutations:

There is no 'first element' in such an arrangement. Any element can be considered the 'first element'.
The number of circular permutations of a set S with 'n' elements is given by
(n – 1)!

c. Multiset permutations :

It is an ordered arrangement of the 'n' elements of set M in which each element appears the same number of times as in M.
A good example of multiset permutation is the anagrams of a word having repeated letters.
Say a word "W" contains 'n' letters, out of which 3 letters are repeating m₁, m₂ and m₃ times. (Their order is not important).
Then the number of multiset permutations of W is given by.

d. Combinations:

It is also a selection of items from a given set. But unlike permutations, here the order of selection of items does not matter.
If the set has 'n' elements, the number of ways of selecting 'r' elements is given by
ⁿC_r =

iv. Number theory:
(i) n! is divisible by all prime numbers upto 'n' (including 'n'). (ii) Consider the sum of reciprocals of factorials. It is a convergent series whose sum is e, the Euler's number. Since 'e' = 2.718281... is an irrational number, the sum of the above series is an irrational number. By multiplying the factorials with positive integers, we can have a convergent series whose sum is rational. By multiplying with (n + 2), we get In the beginning, we said factorial of a negative numbers and fractions is not defined. In advanced mathematics, however, factorial function is extended to non-integer values (both positive and negative).
Double factorial:
The double factorial of an odd positive integer 'n' is the product of all odd integers upto it. It is denoted by n!!. Ex : 11!! = 1 × 3 × 5 × 7 × 9 × 11
Super factorial:
The super factorial (sf) of 'n' is equal to the product of the factorials upto 'n'. Ex : sf(4) = 1! × 2! × 3! × 4! = 288

Applications of factorial (contd..)

ii. Probability:

In the poker card game, the "poker hand" can be described as 5-combinations (r = 5) from a card deck (n = 52). All the 5 cards of the 'hand' are distinct and their order does not matter. So the no. of such combinations is

So the probability (or chance) of drawing any 'hand' at random is 1/25,98,960.
It is a miniscule number. So the probability is very low.

iii. Exponential function:

In the complex plane, the exponential function is defined by the following power series :

When z = it where 't' is real,

In calculus, the Taylor series for the exponential function e^x (at a = 0) is given by

Note:

(2n)! = 2ⁿ . n![1 . 3 . 5 . . . . (2n – 1)]

n! – (n – 1)! = ?

Show that 20! – 19! = 19 × 19! and find its value.

20! – 19!	=	19 × 19!
20! – 19!	=	20 × 19! – 19!
	=	19! × (20 – 1)
	=	19! × 19
	=	19 × 19!
In general, n! – (n – 1)! = n × n!
OR
n! – (n – 1)!	=	n! – (n.(n – 1)!)/n
	=	n! – (n!/n)
In terms of absolute values
20! – 19!	=	20.19.18. . . . 3.2.1 – 19.18.17. . . . 3.2.1
	=	2432902008176640000 – 121645100408832000
	=	2311256907767808000

Gamma function

It is an extension of the factorial function with its argument lowered (or shifted down) by 1.
It is applicable to both real and complex numbers. The gamma function is represented by the symbol Γ.
So if 'n' is a positive integer,

The best known value of the gamma function (without going into details) is

It follows that

Example:

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