Probability > How to Solve Permutations and Combinations Problems
Watch the video or read the article below:
- What are Permutations and Combinations?
- How to Solve Permutations and Combinations Problems by Hand
- Permutations and Combinations in Excel
Permutations are a set of ordered objects. The word “combinations” has slipped into English usage for things like a “combination lock”. The kind of lock you put around your bicycle should be called a “permutation lock,” because the order does matter.
If you don’t care what order you have things, it’s a combination. If you do care, it’s a permutation. Lottery tickets where you pick a few numbers are a combination. That’s because the order doesn’t matter (but the numbers you select do). Picking winners for a first, second and third place raffle is a permutation, because the order matters.
Allowing repetition depends on your situation. For example:
- Combination locks can have any number in any position (for example, 9,8,9,2), so repetitions are allowed. The number “9” appears twice here.
- Lottery numbers don’t allow repetition. The same number won’t appear twice in the same ticket. For example, you can pick numbers 67, 76, and 99. But you can’t choose 67, 67, and 67 as your winning ticket.
Logic should tell you if repetitions are allowed. If you’re dealing with items that aren’t going to be replaced (like lottery balls), then you’re looking at no repetitions allowed.
For repetitions, the formula is nr. N is the number of things you are choosing from and r is the number of items. For example, let’s say you are choosing 3 numbers for a combination lock that has 10 numbers (0 to 9). Your permutations would be 10r = 1,000.
For NO repetitions, the formula is n!/(n-r)!. “!” is a factorial. For example, let’s say you have 16 people to pick from for a 3-person committee. The number of possible permutations is:
16! / (16-3)! = 16! / 13! = 3,360.
The hardest part about solving permutations and combinations problems is: which is which?
Combinations sounds familiar: think of combining ingredients, or musical chords. With combinations, order doesn’t matter: Flour, salt and water in a bowl is the same as salt, water and flour.
Permutation isn’t a word you use in everyday language. It’s the more complex of the two. Every little detail matters. Eggs first? Then salt? Or flour first?
Combinations and permutations each have their own formula:
This is just multiplication and division. The “!” is the factorial symbol. That’s just a special way of multiplying numbers. To get a factorial, multiply the number by each number below it until you get to 1. For example:
4! = 4 x 3 x 2 x 1 = 24
2! = 2 x 1 = 2
Google can work out factorials for you. Type 4! into a Google search and you’ll get the answer (24).
Permutations and Combinations: Sample Problems
Sample problem #1: Five bingo numbers are being picked from a ball containing 100 bingo numbers. How many possible ways are there for picking different numbers?
Step 1: Figure out if you have permutations or combinations. Order doesn’t matter in Bingo. Or for that matter, most lottery games. As order doesn’t matter, it’s a combination.
Step 2: Put your numbers into the formula. The number of items (Bingo numbers) is “n.” And “k” is the number of items you want to put in order. You have 100 Bingo numbers and are picking 5 at a time, so:
Step 3: Solve:
Sample problem #2. Five people are being selected for president, vice president, CEO, and secretary. The president will be chosen first, followed by the other three positions. How many different ways can the positions be filled?
Step 1: Figure out if you have permutations or combinations. You can’t just throw people into these positions; They are selected in a particular order for particular jobs. Therefore, it’s a permutations problem.
Step 2: Put your numbers into the formula. There are five people who you can put on the committee. Only four positions are available. Therefore “n” (the number of items you have to choose from) is 5, and “k” (the number of available slots) is 4:
Step 3: Solve:
Note: Oddly enough, a combination lock has the wrong name. It should be a permutation lock. Why? Because the order matters. Try entering the numbers in the wrong order and see if the lock opens :)
Permutations formula: PERMUT(number, number_chosen)
For example, if you had 100 items and wanted to choose 4, you would type the following into a blank cell:
Combinations formula: COMBIN(number, number_chosen)
For example, if you had 99 items and wanted to choose 10, you would type the following into a blank cell:
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