Combinations
A combination is a way of choosing items from a set, where order does not matter. For example, if you have 5 fruits - apple, banana, cherry, orange, and pear - and you want to choose 3 of them, there are 10 possible combinations:
- Apple, banana, cherry
- Apple, banana, orange
- Apple, banana, pear
- Apple, cherry, orange
- Apple, cherry, pear
- Apple, orange, pear
- Banana, cherry, orange
- Banana, cherry, pear
- Banana, orange, pear
Notice that the order in which we list the fruits doesn't affect the combination. So, the combination of apple, banana, and cherry is the same as the combination of banana, cherry, and apple.
Permutations
A permutation is a way of choosing items from a set, where order matters. Using the same set of 5 fruits, there are 60 possible permutations of 3 fruits:
- Apple, banana, cherry
- Apple, banana, orange
- Apple, banana, pear
- Apple, cherry, orange
- Apple, cherry, pear
- Apple, orange, pear
- Banana, apple, cherry
- Banana, apple, orange
- Banana, apple, pear
- Banana, cherry, orange
- Banana, cherry, pear
- Banana, orange, pear
- Cherry, apple, banana
- Cherry, apple, orange
- Cherry, apple, pear
- Cherry, banana, orange
- Cherry, banana, pear
- Cherry, orange, pear
- Orange, apple, banana
- Orange, apple, cherry
- Orange, apple, pear
- Orange, banana, cherry
- Orange, banana, pear
- Orange, cherry, pear
- Pear, apple, banana
- Pear, apple, cherry
- Pear, apple, orange
- Pear, banana, cherry
- Pear, banana, orange
- Pear, cherry, orange
As you can see, the number of permutations grows rapidly as the number of items increases.
Practice Problems
- A bag contains 5 red balls, 7 blue balls, and 3 green balls. If a ball is drawn at random from the bag, what is the probability that it is blue?
- A coin is flipped 3 times. What is the probability that it lands on heads at least once?
- A deck of 52 cards contains 26 black cards and 26 red cards. A card is drawn at random from the deck. What is the probability that it is a heart?
Tags: math-is-fun, combinations, permutations