Why counting comes first
Games of chance are games with a built-in house edge, and nothing in this article changes that. No strategy, system or timing alters the odds, most players lose money over time, and nobody should gamble money they cannot afford to lose. The counting methods below are general math tools, and the examples use books, locks and committees rather than any real game. Any figures are hypothetical.
Many probability problems start with the same question: how many different outcomes are there? If every outcome is equally likely, the chance of a particular result is the number of ways it can happen divided by the total number of outcomes. Getting those counts right is the hard part, and that is what permutations and combinations are for.
The one question that separates them is simple: does order matter?
Permutations: order matters
A permutation is an arrangement. Swapping two items makes a different permutation.
Say you have 4 different books and want to line them up on a shelf. You have 4 choices for the first spot, then 3 for the second, 2 for the third and 1 for the last:
4 × 3 × 2 × 1 = 24
That product is written 4! and read “four factorial”. Five books give 5! = 120 arrangements, and ten books give 3,628,800. The count balloons quickly.
Often you arrange only some of the items. If you pick 3 of 5 different books and place them in order, you have 5 choices, then 4, then 3:
5 × 4 × 3 = 60
So there are 60 ordered ways to pick 3 books from 5.
Combinations: order does not matter
A combination is a selection. Only which items you picked matters, not the order in which you picked them.
Take the same 3 books out of 5, but now you just want to know which three end up in your bag. Each group of 3 books can be arranged in 3! = 6 orders, and all six of those orders count as the same bag. So you divide the 60 ordered ways by 6:
60 ÷ 6 = 10
There are 10 combinations. The general idea is worth remembering: a combination count is a permutation count divided by the number of ways to reorder the chosen items.
Here is a second check. How many ways can a committee of 3 be chosen from 10 people? Ordered, it is 10 × 9 × 8 = 720. Dividing by 6 gives 120 different committees.
A quick way to decide which one you need
Ask whether rearranging the same items gives you a different result.
- Seating guests in numbered chairs: yes, order matters, so it is a permutation.
- Choosing which guests are invited: no, the same group is the same group, so it is a combination.
- A PIN code of 1-2-3-4 versus 4-3-2-1: order matters.
- A hand of cards dealt from a deck: order usually does not matter, so it is a combination.
Everyday language is unhelpful here. People say “combination lock”, but because the order of the digits matters, a lock is really a permutation-type problem.
When items can repeat
The formulas above assume each item can be used once. A four-digit PIN is different, because digits can repeat. Each of the 4 positions has 10 possible digits, so the count is:
10 × 10 × 10 × 10 = 10,000
When repeats are allowed and order matters, you simply multiply the number of choices at each position. This kind of counting is useful to illustrate how many outcomes a simple system can have. A hypothetical machine with 3 reels, each showing one of 5 symbols, would have 5 × 5 × 5 = 125 possible symbol combinations on the display. A real machine’s design is more complicated, and the outcomes are usually not equally likely, which is a matter of reel strip weighting. The point here is only the arithmetic.
How big can the numbers get?
Combinations are the reason that “pick some numbers from a larger set” problems have such huge totals. Choosing 6 numbers from 49, where order does not matter, gives 13,983,816 possible selections. That is about 14 million, so any one specific selection has a chance of 1 in 13,983,816, or roughly 0.000007%.
A standard 52-card deck gives 2,598,960 different five-card hands, also ignoring order. Those are standard, well-known counts, and they show how quickly possibilities pile up from small starting sets. Seeing figures like these can help put tiny probabilities in perspective.
From counts to probabilities
Once you can count, probability is a division. Suppose you pick 3 books from 5 at random and want to know the chance that a particular book is in your bag. How many of the 10 possible bags contain it? If that book is fixed in the bag, you choose the other 2 from the remaining 4. That is 4 × 3 = 12 ordered ways, divided by 2 for order, giving 6 combinations. The chance is 6 out of 10, or 60%.
The same logic powers many probability calculations: count the successful outcomes, count all outcomes, and divide. When you need the chance of several steps happening together, compound probability takes over from there.
Common mistakes
- Using permutations when order does not matter, which overcounts.
- Forgetting whether repeats are allowed.
- Assuming all outcomes are equally likely. A weighted die breaks that assumption.
Keeping perspective
Counting outcomes is a way to understand chance, not to beat it. A huge number of possibilities does not make any one of them more or less likely to occur next, and it does not create a pattern to exploit. If gambling stops being fun, if you are chasing losses or you are spending more than you planned, consider taking a break and contacting a national problem-gambling helpline or local health service.