Lesson Worksheet: Counting With Repetition Where Order Does Not Matter Mathematics

In this worksheet, we will learn how to use combinations to count with repetition where order does not matter.


A magician wants to hide 4 identical balls in 16 boxes. More than one ball can be placed in a box. In how many ways can the balls be hidden?


Twenty passengers get on an airport shuttle. The shuttle route includes six hotels, and each passenger gets off the shuttle at his/her hotel. The driver records how many passengers leave the shuttle at each hotel. How many different possibilities exist?


How many integer solutions are there for the equation 𝑥+𝑥+𝑥=30, given that 𝑥>5, 𝑥4, and 𝑥>3?


How many possible diagonals can be drawn in a regular 𝑛-gon?

  • A𝑛(𝑛3)2
  • B𝑛(𝑛1)2
  • C𝑛(𝑛1)
  • D𝑛2
  • E𝑛(𝑛3)


Three friends are in a tapas restaurant. They would like to order 6 dishes to share. There are 15 different options on the menu. Given that they can choose multiple dishes of the same type, how many different possible ways can they order 6 dishes to share?


How many nonnegative integer solutions are there to the equation 𝑥+𝑥+𝑥+𝑥+𝑥=40?


Mia is buying an ice cream cone and is able to pick two scoops from four flavors: chocolate, banana, strawberry, and vanilla. How many different ways can she pick two scoops from the four flavors?


A boy is asked to select two flags for football clubs out of 4 different flags. In how many ways can such a selection be made if he can choose the same flag twice and the order of the two flags is not important?


If you need to distribute 5 identical chocolate bars among 3 girls such that each girl takes at least one, in how many ways can you distribute the chocolate bars?


You need to select three letters. The letters will be selected from a set of 4 letters. Repetition of letters is allowed and the order is not important.

How many such selections can be made?

  • A4 selections
  • B12 selections
  • C24 selections
  • D20 selections
  • E3 selections

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