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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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