– English Lyrics

Lyrics & Translation

[English]

This we all know is a standard factorial

where we put an exclamation mark in

front of a number like four factorial

equals four times three times 2 * 1

which is equal to 24 and any n factorial

equals n * n - 1 and so on * 2 * 1. But

do you know that we have other types of

factorials as well and today we will

explore almost all of them. double

factorial. We represent it using a

number followed by a double exclamation

mark in front of it. For an odd number

like 9, it's the product of all odd

numbers from 9 down to 1. So we multiply

9 * 7 * 5 * 3 * 1. Then in the case of

an even number like 8 double factorial

equals 8 * 6 * 4 * 2. Then if we keep

extending this idea we get a

multiffactorial.

Here instead of decreasing by 1 or two

like in single or double factorials we

decrease by a fixed number k. So an n

multiffactorial of order k means we

multiply n * n - k * n - 2 * k and so on

until we reach a number that is greater

than or equal to 1. But the next step

would go below 1. For example, 7

multiffactorial of order 3 would be 7 *

4 * 1. Next up we have rising factorial

which we denote like this. It means we

multiply x * x + 1 * x + 2 and so on

until we do this n times in total. So

this last term will be x + n -1 because

remember we started from zero here and

not one. So this will be n terms. For

example, if we take the rising factorial

of three with subscript four, it will be

three times four * 5 * 6. It is called

rising because the numbers keep

increasing in each step. Similarly, we

have falling factorial which is just the

opposite of rising factorial. It means

we multiply x * x -1 * x - 2 and so on

until we do this n times in total. So

the last term will be x - n + 1 because

again we are taking n terms starting

from x and going down. For example, if

we take the falling factorial of six

with subscript 3, it will be 6 * 5 * 4.

It is called falling because the numbers

keep decreasing with each step. Next up

is super factorial which sounds quite

powerful and it actually is. There are a

couple of ways people define it, but one

common version is where we take the

factorials of all numbers from one up to

n and multiply them all together. We

denote it with a number n followed by a

dollar sign. So a super factorial of 4

would be 1 factorial * 2 factorial * 3

factorial * 4 factorial that is 1 * 2 *

6 * 24 which gives 288.

It grows really fast compared to the

regular factorial. Then we have

factorial tetration. Factorial titration

is when we take a number n first

calculate its factorial and then perform

tetration using that result like n

factorial titration n factorial.

Tetration means repeated exponentiation

just like multiplication is repeated

addition and exponentiation is repeated

multiplication.

For example, three titration four means

3 raised to the power of 3 raised to the

power of 3 raised to the power of three

with four threes stacked like a tower.

Now if we take just n= 3 then 3

factorial is 6 and factorial titration

becomes 6 raised to the power of 6

raised to the power of 6 and so on six

times. This number is far far larger

than 10 raised to the power 80 which is

the estimated number of atoms in the

entire observable universe. Next up we

have hyperfactorials.

Instead of just multiplying numbers like

1 * 2 * 3 and so on. In a hyperfactorial

we raise each number to the power of

itself before multiplying. So for

example a hyperfactorial of 3 would be 1

to the power 1 * 2 ^ 2 * 3 to the^ 3

that is 1 * 4 * 27 which equals 108.

Let me know in the comments what will be

the hyperfactorial of four. Then we have

a gamma function which is a continuous

extension of factorial.

While factorial is only defined for

whole numbers, the gamma function or

this integral lets us calculate

something like factorial for decimal and

even complex numbers. It plays a huge

role in higher mathematics, especially

in calculus, probability and complex

analysis. One of the most fascinating

results of the gamma function is when we

plug in 1/2. Gamma of 1/2 is equal to

the square roo<unk> of pi. That's right.

Gamma of 1/2 equals roo<unk> pi, which

connects factorials to geometry in a

very surprising way. Then we have a

primordial. This means we multiply all

prime numbers up to n and we denote it

using this hash sign. For example, the

primes up to seven are 2, 3, 5, and 7.

So we do 2 * 3 * 5 * 7 to get primal of

7. Noise. Now this is interesting. Look

at it carefully. We have n with an

exclamation mark but it is not after n.

Instead it is placed as a prefix. This

is called a subfactorial which counts

how many ways we can arrange n items. So

none of them are in their original

spots. Like suppose we have three

letters A, B and C like this. Can you

write all the possible ways to arrange

these three letters? First we can write

A, B and C. Then keep A fixed and swap B

and C to get A, C and B. Now let's fix B

at the beginning. After that we can

arrange the remaining two letters A and

C in two ways B A and C and then B C and

A. Finally let's fix C at the beginning

and arrange the rest. That gives us C

then A and B and finally C, B and A. So

in total we have six different

arrangements of these three letters.

This is where factorial comes into the

picture. Since we are arranging three

different letters, we calculate the

total number of arrangements using three

factorial which gives us three * 2 * 1

or six. And that's exactly the number of

different arrangements we got. Now in

the case of a subfactorial, we count how

many ways we can arrange A, B and C so

that none of them are in their original

spots. For that suppose the initial

arrangement is A, B and C. Now from

these six possible arrangements remove

all the A from the first position which

means this and this will get cancelled

out. Now remove all the B from the

second position which means this and

this will get cancelled out. Now remove

all the C from the third position which

means this and this will get cancelled

out. So these four arrangements do not

count in case of subfactorial because

now none of them are in their original

position when compared with this initial

arrangement. This is the way to

calculate a subfactorial.

In case of three subfactorial, write 1 /

0 factorial minus 1 over 1 factorial + 1

/2 factorial - 1 over 3 factorial. Now

we solve each term. 1 / 0 factorial is

1. 1 over 1 factorial is 1. 1 / 2

factorial is 1 / 2. 1 over 3 factorial

is 1 / 6. 1 - 1 is 0. Then 1 / 2 - 1 / 6

gives us 1 / 3. After that we multiply

the answer with 3 factorial which is 6.

So we do 6 * 1 / 3 which gives 2. That

means the sub factorial of three is 2.

This tells us that there are two ways to

arrange three items in such a way that

none of them remains in their original

place. This kind of arrangement is

called a derangement. Okay, let me know

in the comments what will be the

subfactorial of seven. Then we have

exponential factorial. Instead of

multiplying numbers or raising them to

themselves, here we build a power tower

starting from 1 up to n. So an

exponential factorial of four would be 1

raised to the power 2 raised to the

power 3 raised to the power 4. Easy

peasy. Finally, we have quantum

factorial which comes from the world of

quantum mathematics and is used in areas

like quantum groups and Q calculus.

Instead of regular numbers, we deal with

what's called Q numbers. A Q number for

any N is written as open bracket N close

bracket with a small q and it is equal

to 1 minus q to the power n / 1 - q. So

the quantum factorial of n is the

product of all q numbers from 1 to n.

For example, if n is 3 and q is 2, then

open bracket 1 close bracket q is 1,

open bracket 2 close bracket q is 3 and

open bracket 3 close bracket q is 7. So

the quantum factorial becomes 1 * 3 * 7

which is 21. It's like a flexible

version of factorials that depends on

the value of Q. And when Q gets close to

one, the quantum factorial becomes the

same as the usual factorial. So goo.

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