What is a factorial?

1 – Five books to read … but in what order? factorial

You are about to go on vacation … factorial

As you like to read novels, you are about to choose five to devour at night, comfortably installed in a quiet place. Once the books are selected, a question remains: in what order are you going to read them?

To simplify things, let’s look at each of these five books by its color:

PICT

You can start with red, then continue with yellow, purple, green and finally blue.

But you could just as well start with yellow, follow with blue, then red, green and finish with purple.

There are, it seems, many possibilities … but how much?

The answer is “factorial 5”, that is to say 5\times4\times3\times2\times1 is 120 opportunities.

It’s pretty easy to understand: the first book you’ll read can be chosen in 5 ways. For each such choice, the second book can be chosen in 4 ways and therefore exists 5\times4=20 ways to choose the first two books. The third book can then be chosen in 3 ways and we are at 5\times4\times3=60 possible scenarios for the first three books. Then comes the fourth book, which can be chosen in 2 ways and the fifth and last one that is imposed!

2 – The factorial of an integer

More generally, if n is an integer greater than or equal to 1, we denote by “factorial n »The product of the integers of 1 at n. This integer is noted \boxed{n!} and the table below shows the values of the first terms of this sequence:

PICT

The product of the integers of 1 at n+1 is clearly equal to the product of n+1 by the product of the integers of 1 at n. In other words, we have for everything n\geqslant1 :

    \[ \boxed{\left(n+1\right)!=\left(n+1\right)\: n!}\]

We can define the factorial of 0 by ensuring that this recurrence formula is respected for n=0, which leads to ask:

    \[ \boxed{0!=1}\]

By generalizing what has been said above about the 5 books, we see that the whole n! is interpreted as the number of ways to swap n objects. factorial

3 – Permutations of a card game

Given a deck of 52 cards, the number of ways to order them is 52!factorial

This is an absolutely colossal whole. Here it is, in flesh and bone:

    \[ 52!=80658175170943878571660636856403766975289505440883277824000000000000\]

is around 0,8\times10^{68}.

So here is a story that takes place in 52! seconds (sit comfortably: it will last a moment) … factorial

One guy walks along the Earth’s equator (about 40\thinspace000\thinspace km) at the rate of one step every billion years. factorial

pict

If he crosses a meter with each step, he will not need less than 40\thinspace000\thinspace000\thinspace000\thinspace000\thinspace000 years to go around the world (a duration VERY much higher than the age of the universe … but good). factorial

When he has finished touring the globe, he takes the pipette he has in his pocket and takes a drop of water in the Pacific Ocean, then leaves for the next round, always at the same rate of a not every billion years. At the end of the second round, he takes a second drop with his pipette and so on … factorial

PICT

I was told that the Pacific Ocean would contain about 700\thinspace000\thinspace000\thinspace km^{3} of water. Moreover, we can reasonably accept that a cubic centimeter of water can form 20 drops … It will take “a certain time” (as Fernand Reynaud would have said) to our character to empty the ocean! But he will eventually get there.

Incidentally, one wonders where he will store all the water he extracts … but the story does not say … and we’re not close to that

So when the ocean is empty, our friend pours all the water (!) And places a sheet of paper on the table in front of him. factorial

PICT

And he starts again: a trip around the world … a drop … a trip around the world … a drop … etc …

Each time the ocean is empty, it fills it again, then places a new sheet of paper on the pile. factorial

The thickness of the sheet pile therefore increases gradually and, after a very very long time, it will eventually reach the value of the distance between the earth and the sun, which is approximately 150\thinspace000\thinspace000\thinspace km. factorial

And here you say it’s good, we’re done! Well no, because at that moment, our character must take the whole process 3000 time !! factorial

And it’s only at the end of the 3000– step that will have passed (roughly) 52! seconds. factorial

You do not believe in it ? Admittedly, it’s hard to believe, but the facts are stubborn: factorial

The time required for a world tour is 4\times10^{16} years, ie (in seconds):

    \[ T=4\times10^{16}\times365\times24\times3600=1\thinspace261\thinspace440\thinspace000\thinspace\times10^{15}\thinspace s\]

As the ocean encloses:

    \[ N=700\thinspace000\thinspace000\times100\thinspace000^{3}\times20\thinspace\text{ drops of water}\]

it will be necessary to wait:

    \[ N\times T=1766016\times10^{43}\thinspace s\]

Then, assuming that the sheets of paper have a thickness of a tenth of a millimeter, it will take in all:

    \[ \frac{150\thinspace000\thinspace000\thinspace000\thinspace000}{0,1}\:\text{leaves}\]

so that the battery has the desired thickness. In total, the time required for the entire process is:

    \[ D=3000\times1766016\times10^{43}\times1\thinspace500\thinspace000\thinspace000\thinspace000\thinspace000\thinspace s\]

is :

    \[ D\simeq0,8\times10^{68}\thinspace s\]

We find a number of the same order of magnitude as 52! factorial

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