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The Counting Principle

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The basic counting principle states that if there are X ways to do one thing, and Y ways to do another, then there are X × Y ways of doing both.

Similarly, if there are X ways to do one thing, and Y ways to do second one and Z ways to do third one, then there are X × Y × Z ways of doing all the three.

Example: There are 6 pink kites and 4 yellow kites. Daniel wants to choose 2 kites – 1 pink and the other of yellow color. In how many ways can Daniel choose the kites?

The Counting Principle states that we multiply the number of choices to find the total number of possibilities.

So, number of ways Daniel can choose the kites = 6 × 4 = 24 ways

Example: Isabelle wants to order frozen yoghurt. If there are 5 different flavours of yoghurt and 14 different toppings to choose from, then in how many different ways can Isabelle place her order?

Number of ways Isabelle can place her order = 5 × 14 = 70 ways


Check Point

  1. Sandy wants to buy flowers for his friend. There are 5 colors of roses, 4 colors of lilies, and 2 types of orchids. In how many ways can Sandy choose 3 flowers of each type?
  2. A kitchen garden shop has 6 types of floral plants and 12 plants of different vegetables. Hebrew wants to buy one floral plant and one plant from vegetables. In how many ways can he choose the 2 plants?
  3. Shawna’s doll has 6 different dresses, 5 sandals and 4 colors of hairs, which are detachable and can be replaced. In how many ways can Shawna dress up her doll?
  4. Millie’s grandmother bought 2 types of bread and made 4 types of fillings for sandwiches. In how many different ways can her grandmother make sandwiches?
  5. Peter has 6 shirts, 3 neck ties and 4 trousers for a party. In how many ways can he dress up himself?
Answer key
  1. 40 ways
  2. 72 ways
  3. 120 ways
  4. 8 ways
  5. 72 ways
Download/Solve a Worksheet for The Counting Principle

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