Understanding why entropy must always increase is not too difficult. There are many more ways to arrange items so that they are in disarray than there are to arrange them in a nice orderly way. So, given free reign, things tend to get disorderly. As an example, consider a deck of cards.
It is possible to arrange a deck of cards so that all the suits are together and the cards are all in order, from ace to king. But, the number of ways to do so is pretty small. We could arrange them such that the spades were on top, followed by the hearts, then the clubs, and then the diamonds. Or, we could start with the clubs, then have the diamonds, clubs, and spades. Since this system is so simple, it is pretty simple to calculate how many ways we can arrange the deck such that all the suits are collected together and the cards are ordered. We have four suits, so there are four choices for the suit we want to use first. For each of those choices, there are three remaining choices of which suit to place second. So, that is 4 × 3 = 12 choices of which suits to put in first and second place. For each of those choices, we have two remaining choices for the third position because there are only two suits left to be placed, so we have 4 × 3 × 2 = 24 possibilities for the placement of the first three suits.
The final suit does not give us any more choices. We have already placed three of the four suits, so only one remains. Thus, the total number of possible ways we can stack a deck of cards such that all the suits are collected and the individual suits are ordered from ace to king is 4 × 3 × 2 × 1 = 24.
You can come up with other ways to order the cards, perhaps collecting all the aces together, and all the twos, threes, etc. You could go through a calculation similar to the one above to figure out how many ways the cards can be collected that way. Do you think you would have more ways, or fewer? If you like to play poker, then you might already be familiar with these sorts of calculations.
But, what if you want to arrange the cards in no particular order at all? How many ways are there to do that? Well, for the first card, we can choose any of the 52 in the deck (we are ignoring the jokers). For the second, we have one less, so there are 51 possibilities from which to choose. So, for the first two cards, we have 52 × 51 = 2652. We already have many more ways to place the first two cards than we had to place all four suits in the first example. How many choices do we have for placing a third card? Fifty, because that is how many cards are left to place. You can probably see the pattern here. To place all the cards, we will have a number of possibilities given by 52 × 51 × 50 × 49 × 48 × … × 5 × 4 × 3 × 2 × 1. We simply take one off the previous number of possibilities and multiply by that, repeating this pattern until we get down to one. Such a pattern, as you might already know, has a name. It is called the factorial function. We could shorten our calculation for the cards by saying that the number of ways of arranging the 52 cards in the deck is 52!, where the exclamation point means to take the factorial of 52. This is a huge number. It is almost 1068.
Now imagine that instead of placing cards from a deck of cards, we are placing gas molecules in a box. Each has a position and a velocity. We can choose the x, y, and z components of the position and the x, y, and z components of the velocity independently. How many different ways are there to do that if the box contains a mole (6.02 × 2023) of atoms? We will not compute it, but you probably see that the number is enormous, much larger than the ways we can place 52 cards. The analysis for molecules has to be slightly modified because, unlike cards in a deck, molecules are identical: every oxygen molecule is like every other oxygen molecule, every nitrogen is like every other nitrogen, and so on for carbon dioxide, etc. We have to take this notion into account, but it does not change the basic analysis. There are vastly more ways to arrange things such that they are disordered rather than ordered.
The second law of thermodynamics makes a very strong statement about this idea, saying that any change in the Universe always increases the total entropy of the Universe. Of course, physicists have a very precise mathematical expression to define what is meant by entropy, but we do not have to consider the details to that level. We should mention that an overall increase in “disorder” or “disarray” in the universe does not prohibit the growth of complex structures.