Worksheet: Counting Using Combinations

In this worksheet, we will learn how to use combinations to solve counting problems.


How many 3-card hands can be chosen from a deck of 52 cards?


Determine the number of ways to choose 3 people from 55.


In how many di๏ฌ€erent ways can 12 people be picked from 15?


In a class, there are 18 boys and 7 girls. Determine the number of ways that a team of 3members can be selected.


At a college, there were 560 ways to select 13 students to attend a seminar. Determine the number of students at the college.


Given 87 points arranged in a circle, determine the number of line segments which can be formed using these points.


Victoria was asked by her teacher to choose 5 from the 8 topics given to her. How many different-five-topics groups could she choose?


The names of 4 students are each written on a piece of paper which are then placed in a hat. If 2 names are randomly selected from the hat, determine the number of all two-student selections that are possible.


Michael attended a meeting at his firm. If each of the 11 attendees shook hands once with each of the other attendees, how many handshakes were there?


Ameliaโ€™s teacher divided the class into groups of six and required each member of a group to meet with every other member of the same group. How many meetings will each group have?


A village has 2 committees, each containing 2 people. In how many ways can the committees be formed if the members are selected from 12 people with the condition that a person can only be chosen once?


A class contains 14 boys and 13 girls. In how many ways can you select a team of 8 people from the class such that every member of the team is of the same sex?


A chess tournament is to be held, where every player plays each of his opponents. Given that there are 78 participants, calculate the number of games that will be played.


There are 7 red balls and 6 white balls in a bag. Determine the number of ways of selecting 4 red balls and 3 white balls.


In a football league of 8 teams, each team plays 2 matches against every other team. How many football matches are played in a season?


In how many ways can a teacher choose one or more students out of a class of 9?


If the elements of the set {๐ด,๐ต,๐ถ,๐ท} are arranged on a circle such that no two elements coincide, determine the number of polygons which can be made using these points as vertices.


Let ๐‘‹={๐‘ฅโˆถ๐‘ฅโˆˆโ„ค,10โ‰ค๐‘ฅโ‰ค16} and ๐‘Œ={{๐‘Ž,๐‘}โˆถ๐‘Ž,๐‘โˆˆ๐‘‹,๐‘Žโ‰ ๐‘}. Determine the value of ๐‘›(๐‘Œ), where ๐‘›(๐‘Œ) is the number of elements in ๐‘Œ.


When installing a combination lock, you must choose four different digits from 0โ€“9 to form the key. The order of the digits is not important, so โ€œ1,234โ€ and โ€œ2,134โ€ represent the same key.

How many different keys can be programmed into the lock?

In how many different ways can a key be entered?

A company using this lock decides they want a more secure choice. They opt for a lock where the order in which the four different digits are entered matters. How many keys could be programmed into this lock?


In how many ways can 34 students be selected from 36?


If 6 people apply for 5 vacancies at a law firm, determine the number of ways the firm can fill the vacancies.


Determine the number of ways to choose 10 different letters, or 6 different letters, from a choice of 21 different letters.

  • A๏Šจ๏Šง๏Šง๏Šฆ๏Šจ๏Šง๏Šฌ๐‘ƒร—๐‘ƒ
  • B๏Šจ๏Šง๏Šง๏Šฆ๏Šจ๏Šง๏Šฌ๐ถร—๐ถ
  • C๏Šจ๏Šง๏Šง๏Šฆ๏Šจ๏Šง๏Šฌ๐ถ+๐ถ
  • D๏Šจ๏Šง๏Šง๏Šฆ๏Šจ๏Šง๏Šฌ๐‘ƒ+๐‘ƒ


How many ways can a baseball coach arrange the order of 9 batters if there are 15 players on the team?


How many unique ways can a string of Christmas lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs?

  • A9,720
  • B155,520
  • C6,480
  • D4.83ร—10๏Šฎ
  • E3.03ร—10๏Šง๏Šฏ


A motorcycle shop has the following vintage motorcycles: 10 choppers, 6 bobbers, and 5 cafe racers. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 cafe racers for a weekend showcase given that all the motorcycles of each type are unique?

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