If we shuffle a deck of cards, it’s likely that the exact order we put them in has never existed before in the history of the universe.
Although our school memory of Combinatorics may not be very happy (that of variations, permutations, and combinations), we must recognize that it provides very useful tools, both in our daily lives and in matters of greater significance. Practically everyone (who is currently not engaged in anything related to mathematics) will blurt out something like: “I never understood it”, “I couldn’t tell when they were variations or combinations”, “It was very complicated”. .. But, in hindsight, let’s be honest: why didn’t we understand it? Why didn’t we like it?
The answer to the second question is simple: we didn’t like having to make an effort, think (surely not now, who likes it?). The human being tends to the comfortable, to the easy, and gets used to it quickly.
We are lazy by nature. Everyone. And surely the one who writes this, one of the most. We preferred to solve an already stated equation than to find it from a “statement problem” , just as we preferred to memorize geographical or historical data and release them like parrots on a piece of paper, to find out the type of climate according to some data, or to reasonably compare the government of Carlos V with respect to that of Felipe II, for example. What if we tried to teach all the subjects in a practical way, with something more than the use of memory and something else? Now, I know what you think: it would increase school failure. And perhaps also the number of critical citizens, which hardly anyone is interested in. Let’s leave it there.
Fortunately, whoever reads these lines no longer has to take an exam and does so out of pure curiosity, pleasure, or the desire to remember, learn or simply read some of that mathematics that everyone now says is so useful (let’s see if I can convince you! !!). As a general idea, keep in mind that variations, permutations, and combinations are tools that allow us to count the number of possibilities that we can find when we work with a concrete and a finite number of objects. But to count them in an intelligent way, without having to waste time. For example, if I want to know how many flags of four colors I can compose with seven colors without repeating any of them, or repeating a maximum of two, wasting time would be to make them all and then count how many there are. In addition, so we will always have the reasonable doubt of, have I repeated any? or will they all be? And above with how lazy we said we were at the beginning, to trust.
Obviously, having the help of these tools is much more advantageous the greater the number that we are going to obtain or the objects with which we work. And it also allows us to compare magnitudes or phenomena that can astonish us, such as the one I am going to propose, which is the final score of all this long introduction that I am giving you.
Avogadro’s number
Let’s park the combinatorics for a moment. We are going to try to understand how other scientists have calculated the number of atoms on Earth, which, as everyone assumes, are many, a huge quantity, but a finite quantity. Let’s see how to estimate such a number.
In 1811, the physicist and chemist Amadeo Avogadro proposed that the volume of a gas, at a specific pressure and temperature, does not depend on the substance that constitutes it, but is only proportional to the number of atoms or molecules. Once the idea was raised, there were different attempts to calculate what constant of proportionality that was, if it existed.
Finally (of course, there is a long story behind it) its value was determined: 6.023 x 10^23. Later, in 1971, with the introduction of the mole as a basic unit, that number became a physical quantity, with units mol^(–1) in the International System, since then it has been called Avogadro’s constant. Its value is then 6.023 x 10^23 molecules/mol. Let’s see with an example what it measures.
Consider a mole of water. We know that water consists of two hydrogen atoms to one oxygen (if we dust off the elementary formulation, that is clear from its chemical expression: H 2 O). If we locate in the periodic table the oxygen atom (atomic number Z = 8) has an approximate weight of 16 grams, and hydrogen (Z = 1) one gram. So a mole of water weighs 18 grams and has 6.023 x 10^23 molecules.
Taking the density of water as the unit, a simple rule of three (apologies to those who hate them, but what do you want me to say, they taught me what little I know about chemistry with them), gives us that a liter of water (10^3 grams), assuming its density is unity, contains
To make the calculation on Earth, something similar is done, except that the composition of the metals it contains must first be taken into account, which is approximately the following: iron, 34.6%; oxygen, 29.54%, silicon, 15.2%; magnesium, 12.7%; nickel, 2.4%; sulfur, 1.9% and titanium, 0.05%. And you also have to consider the percentage of water it has, which is 71% of the earth’s surface. Total,, after taking into account these percentages, we can finally estimate that the ‘mole of Earth’ is 38 grams. Google tells us that the mass of the Earth is estimated to be 5,972 x 10^24 kilograms. So an account like the one above gives us a result
For rounding, let’s say 10^50, that is, a one and fifty zeros. A really big number, no doubt.
A simple deck
However, it is a surmountable number. And it is also in our hands. Let’s take a French deck, you know, the poker one, with its diamonds, clubs, hearts, and spades. It has 52 cards. In how many different ways can we arrange those cards? This is where the combinatorics that I told you about before coming in, and the permutations. A permutation is any change of order that we make to the elements of a set. Assume the initial order of the cards in the deck. The first card is the ace of hearts. In how many different positions can we place the ace of hearts in the deck of cards? Clearly in 52 positions, because there are 52 cards (we can put it first, second, third, etc.). And the second card, the two hearts? Taking into account that we have already considered all the positions in which we can place the ace of hearts, we now have 51 chances left for the deuce of hearts (ie, with the position of the first fixed, there are 51 chances left for the second). Considering both at the same time, we will have the product 52 x 51 possibilities. To understand why they multiply, think of it with only two or three cards. Considering all the cards in the deck, all 52 at once, we find the product
what we mathematicians write for short as 52! (read 52 factorial). That is, the total number of permutations of a set of n elements is n! In the case of the deck of 52 cards, that amount, 52! is
80658175170943878571660636856403766975289505440883277824000000000000
If you take the trouble to count the number of figures that appear, they turn out to be 68, that is, an 8 and 67 zeros. Therefore, of the order of 10^67. Much greater than the number of atoms on Earth, which I remind you were of the order of 10^50.The universe would probably end before we got to a billionth of the possibilities without having found a repeat.
What conclusions can we draw from this value? In addition to the one indicated (that the number of possible configurations of a deck of cards is greater than the number of atoms on Earth), if we shuffle a deck of cards, it is probable that the exact order in which we put them, will never have existed before in the history of the universe, that no one has ever arranged that configuration before. And look what there are and have been gamblers on the face of the Earth!! Because given the immensity of the calculated number of possibilities of arranging the cards, the universe would probably end before we reached a billionth of the possibilities without having found a repetition.
At this point, let’s stop for a moment. Are you telling me that out of the zillions of times a person has played cards, he has never laid out the cards the same way? Surely, playing a certain game, solitaire, for example, we have repeated the same configuration several times, because card games tend to order the deck in a certain way according to their rules and their objectives. That is, the moment we introduce some rules, we lose the randomness. In fact, it has also been estimated that the deck needs to be shuffled about seven times with some of the known techniques (for example, the popular American mix, ‘riffle shuffle’, in English) so that it is perfectly shuffled (tell that to the countless cheaters, in a good way, that swarm around the world! !!).
Of course, the other conclusion, which is the most interesting, is that thanks to a simple mathematical tool I am able to know exactly the total number of different configurations that I could make with a deck of 52 cards (although I would not have enough life to detail them all). ). For the Spanish deck of 40 cards, things are not so spectacular, because
40! =815915283247897734345611269596115894272000000000
In other words, ‘only’ one digit and forty-seven, which does not reach the estimated number of atoms on Earth.
So, when you pick up a deck of cards again, next Christmas, for example, do it with some respect, because you have a gigantic number of possibilities in your hand. And there are games with as many configurations as possible. Do you dare to think about chess, or Go?
Editorial Team
