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Permutations and Combinations: How to Solve Problems

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How to Solve Permutations and Combinations Problems: Overview

The hardest part about solving permutations and combinations problems is: Which is which? Combinations sounds familiar; Think of combining ingredients, or musical chords. Permutation isn’t a word you use in everyday language. With combinations, order doesn’t matter: Flour, salt and water in a bowl is the same as salt, water and flour. Permutations are 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:
permutations and combinations
This is just multiplication and division. The “!” is the factorial symbol. That’s just a special way of multiplying numbers. See: What is a Factorial?

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 number combinations?

Step 1: Figure out if you have permutations or combinations. Order doesn’t matter in Bingo. Or for that matter, most lottery games.

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:
combinations and permutations4

Step 3: Solve:
combinations and permutations3

That’s it!
Sample problem #2. Five people are being selected for president, vice president, CEO, and secretary. 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:
combinations and permutations2

Step 3: Solve:
combinations and permutations

That’s it!

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 :)

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