Probability and combinatorics explained, without memorising

If there is one topic where I see students who are brilliant at calculus go completely to pieces, it is probability. Not because it is hard — indeed, the calculations are often lighter than an integral — but because it is the only maths chapter in high school where applying a procedure is not enough. You have to understand what is happening in the problem, and this throws those who are used to “I find the right formula and apply it”.

I know the result by heart: the student arrives at the lesson with a sheet full of combinatorics formulas (permutations, arrangements, combinations, with and without repetition…) and uses them at random, hoping to hit the right one. Sometimes it works, often it doesn’t, and either way they have no idea why. In this article I do the opposite: I explain the reasoning underneath each formula, so that by the end you can rebuild them instead of remembering them. It is the way I tackle this chapter when I prepare someone for the maturità maths paper (the maths paper of Italy’s school-leaving exam) or for an admission test.

What probability really is

Let’s start from the definition everyone knows and almost nobody uses well. The classical probability of an event is:

It seems trivial, but there is a hidden condition in it that trips up half of students: this formula holds only if all the possible cases are equally likely. Rolling a fair die: yes, each face has probability 1/6. But “tomorrow it rains or it doesn’t, so 1/2” is wrong, because the two outcomes are not equally likely.

The real work, then, is not dividing two numbers. It is counting correctly the favourable cases and the possible cases. And this is exactly where combinatorics comes in: it is not a separate chapter you have to study “on top”, it is the tool you need to calculate those two numbers when there are too many to list by hand.

Keep this in mind: combinatorics = the art of counting without counting one by one. Every time a probability problem seems hard, almost always the difficulty is in the numerator or the denominator of that fraction, not in the probability itself.

The principle that holds everything up: multiplying choices

Before the formulas with complicated names, there is only one principle to really understand, and all the others derive from it. It is called the fundamental counting principle and it is this: if one choice can be made in $m$ ways and a second independent choice in $n$ ways, the two together can be made in $m\times n$ ways.

A concrete example: you have 3 T-shirts and 4 pairs of trousers. How many outfits? $3\times 4=12$. You don’t need a formula, you need to understand that for each T-shirt you have 4 possible pairs of trousers.

Everything comes from this idea. Car number plates, passwords, restaurant menus, the results of several dice rolls: they are all “multiply the choices step by step”. If you train yourself to see problems as a sequence of choices, you have already solved 70% of combinatorics. The formulas with names are only for the recurring cases, so you don’t redo the reasoning from scratch each time.

Permutations, arrangements, combinations: the mental map

Here is the knot where everyone gets lost. There are three words that seem interchangeable and are not. The key to telling them apart is two questions:

1. Does order matter? (Is AB different from BA?)
2. Do I use all the elements or only some?

With these two questions in your pocket, you no longer have to remember which formula to use: you deduce it.

Permutations — order matters, I use them all

A permutation is a way of ordering all the elements of a set. In how many ways can 5 people sit on 5 chairs? The first chair has 5 candidates, the second 4 (one has already sat down), then 3, 2, 1. I multiply:

So $5!=120$. Note that you are just applying the counting principle: the factorial is “multiply the choices that shrink by one”.

Arrangements — order matters, I use only some

If instead I choose and order only $k$ elements out of $n$ (a podium of 3 out of 8 athletes, gold is different from silver): I start from $n$, then $n-1$, and stop after $k$ factors.

Podium with 8 athletes: $8\times 7\times 6=336$. Again, I am not memorising: I am counting the choices one at a time and stopping at $k$.

Combinations — order does NOT matter

This is the most insidious. If I choose $k$ elements out of $n$ but order doesn’t matter (a hand of 5 cards, a committee of 3 people), I have to take the arrangements and remove all the ways the same $k$ elements can be reordered among themselves (that is $k!$):

Drawing 5 cards from 52 (in poker order doesn’t matter):

The trick to never go wrong: if swapping two elements gives you a different configuration, order matters (arrangements); if it gives you the same thing, it doesn’t (combinations). Podium → order matters. Committee → no. Lottery numbers → no. This single question saves you from half of the mistakes in the test.

Conditional probability: when a piece of information changes the cards

Once combinatorics is behind you, the other big hurdle is conditional probability: the probability that $A$ happens given that $B$ has already happened. It is written $P(A|B)$ and the formula is:

The intuitive idea is simple and worth fixing in your mind: when you know that $B$ has happened, your “world of possible cases” shrinks to only the cases in which $B$ is true. You no longer look at the whole universe, you look only at the slice $B$.

Classic example: I draw a card from a deck of 40. What is the probability that it is a king, given that it is a face card? There are 12 face cards, the kings among the face cards are 4, so $4/12=1/3$, not $4/40$. The information “it is a face card” has narrowed the field.

Linked to this is the concept of independent events: two events are independent if the occurrence of one doesn’t change the probability of the other, that is

In that case — and only in that case — it holds that

Careful: two coin tosses are independent; drawing two cards without putting them back in the deck is not, because the first draw changes the deck for the second.

The mistakes I see most often

After many lessons on this chapter, the stumbling blocks are almost always the same five. If you recognise them in advance, you avoid most of the mistakes in the test.

1. Confusing arrangements and combinations. It is mistake number one. Before choosing the formula, stop and ask yourself aloud: “does order matter?”. Write it beside the problem. Thirty seconds that are worth marks.

2. Adding probabilities that are not mutually exclusive.

   $$P(A\cup B)=P(A)+P(B)$$

   holds only if $A$ and $B$ cannot happen together. Otherwise you have to subtract the intersection:

   $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

   With “at least one success” it is often convenient to go through the opposite event (see below).

3. Multiplying probabilities of events that are not independent. The rule

   $$P(A\cap B)=P(A)P(B)$$

   is a shortcut valid only with independence. In draws “without replacement” it is almost always wrong.

4. Forgetting the complementary event. When the problem says “at least one”, calculating directly is a nightmare. The trick is

   $$P(\text{at least one})=1-P(\text{none})$$

   “At least one 6 in four rolls” is done in one line going through the complement, in half a page doing it head-on.

5. Counting the same case twice. It happens when you list the favourable cases “by hand” without a systematic criterion. Better to set up the count with the combinatorial formulas than to keep a list by eye.

How we work on this chapter in lessons

My method here is different from other topics, because probability is not learned by watching: it is learned by trying to count wrongly and understanding why. A typical path:

• First the two questions, then the formulas. In the first lesson we don’t even touch $n!$. We do ten problems deciding only “does order matter? do I use all the elements?”. When that reflex is automatic, the formulas become obvious.
• Rebuild, don’t memorise. I have you derive arrangements and combinations from the counting principle until you no longer need the formula sheet. In the test, if you have a memory blank, you get them back in thirty seconds.
• Tree diagrams for conditional probability. For multi-stage problems (successive draws, urns, diagnostic tests) the tree makes visible what to multiply and what to add. It is the tool that unblocks the largest number of students. If you would rather see it move, the interactive probability tree lets you move the numbers on the branches and watch the products, total probability and reverse reading recalculate.
• Exam-style problems from the start. Whether it is the second paper or an admission test, we work on the real questions of past years, timed, because combinatorics in a test is also a race against the clock.

On timing I am honest: if the basics are there, 3-4 lessons are enough to go from panic to handling combinatorics and basic probability with confidence. To get to the toughest questions of the second paper or the TOLC (the Italian university admission test), you generally need 5-6 meetings with exercises between one lesson and the next. It is not a chapter that takes months — it takes the right reasoning and a bit of targeted training.

Among my students, final-year maths and admission tests are bread and butter: average grade 8.2/10, over 215 students followed and 97% of admission tests passed. Probability is often exactly the chapter that makes the difference between a 7 paper and a 9, because many leave it behind.

If you are preparing for the exam, a probability question comes up practically every year in the second paper: train with the free maturità maths practice tests, which have step-by-step guided solutions. To fit this chapter into the calendar of the final year, there is final year: the plan for the maturità from September.

If you follow the IB, the Applications and Interpretation route gives much more weight to statistics and probability than to calculus: how to choose between the two and the Italian-English glossary are in IB Maths AA or AI. For the combinatorics part on high school questions there is also the Year 4 test on combinatorics and probability.

If probability and combinatorics are your weak point — or if you are preparing for the maturità or a test and want to get there without surprises — write to me for a free introductory call: in half an hour we work out where you are and build the shortest path to where you want to be.