Combinations

Key notes :

Definition of Combinations

  • A combination is a way of selecting items from a group where the order does not matter.
  • Example: Choosing 2 fruits from {Apple, Banana, Orange} → {Apple, Banana}, {Apple, Orange}, {Banana, Orange}.

Difference Between Combinations and Permutations

  • Combinations: Order does not matter. (Example: Choosing 2 team members from a group).
  • Permutations: Order does matter. (Example: Assigning 1st and 2nd place in a race).

Formula for Combinations

  • The formula for finding the number of ways to choose r items from n items is:

C(n,r) = n! / r!(n−r)!

  • Example: Choosing 2 students from a group of 4 →

C(4,2) = 4! / 2!(4−2)! = 4×3 / 2 × 1 = 6

Examples of Combinations in Daily Life

  • Choosing a group of friends for a trip.
  • Selecting different toppings for a pizza.
  • Forming teams in a class project.

Key Concept: Order Does Not Matter

  • If you select A and B, it is the same as selecting B and A.
  • This makes combinations different from arrangements.

Learn with an example

🌀 Dwayne wants to get a sandwich from the deli for lunch. He can pick a roast beef, turkey, or tuna sandwich. The bread choices are white or multi-grain.

How many different combinations can Dwayne pick from?

​ _______ combinations

There are 3 sandwich choices and 2 bread choices. Multiply to find the number of possible combinations:

3 × 2 = 6
There are 6 combinations.

You could also make a list of all the combinations.

Now count the combinations. There are 6 combinations.

🌀Riley is ordering a birthday cake for a friend. The bakery makes carrot, chocolate, and lemon flavoured cakes. Each flavour of cake can come with yellow or green frosting.

How many different combinations can Riley choose from?

​ ________ combinations

There are 3 flavour choices and 2 frosting choices. Multiply to find the number of possible combinations:

3 × 2 = 6
There are 6 combinations.

You could also make a list of all the combinations.

Now count the combinations. There are 6 combinations.

🌀 Krysta is going out for a pasta dinner. She can choose fettuccine, shell, or macaroni pasta, and she can have garlic, cream, or pesto sauce.

How many different combinations can Krysta have for dinner?

​ ________ combinations

There are 3 pasta choices and 3 sauce choices. Multiply to find the number of possible combinations:

3 × 3 = 9
There are 9 combinations.

You could also make a list of all the combinations.

Now count the combinations. There are 9 combinations.

Let’s practice!🖊️