Worksheet: Counting Using Combinations

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

Q1:

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

Q2:

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

Q3:

In how many different ways can 12 people be picked from 15?

Q4:

In a class, there are 42 boys and 39 girls. Determine the number of ways that a team of 3 boys can be selected.

Q5:

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

Q6:

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

Q7:

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

Q8:

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.

Q9:

Fady 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?

Q10:

Dina’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?

Q11:

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?

Q12:

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?

Q13:

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.

Q14:

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.

Q15:

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?

Q16:

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

Q17:

If the elements of the set { 𝐴 , 𝐡 , 𝐢 , 𝐷 } 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.

Q18:

Let 𝑋 = { π‘₯ ∢ π‘₯ ∈ β„€ , 1 0 ≀ π‘₯ ≀ 1 6 } and π‘Œ = { { π‘Ž , 𝑏 } ∢ π‘Ž , 𝑏 ∈ 𝑋 , π‘Ž β‰  𝑏 } . Determine the value of 𝑛 ( π‘Œ ) , where 𝑛 ( π‘Œ ) is the number of elements in π‘Œ .

Q19:

When installing a combination lock, you must choose four different digits from 0–9 to form the key. The order of the digits is not important, so β€œ1,234” and β€œ2,134” 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?

Q20:

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

Q21:

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

Q22:

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

  • A        𝑃 + 𝑃
  • B        𝐢 Γ— 𝐢
  • C        𝑃 Γ— 𝑃
  • D        𝐢 + 𝐢

Q23:

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

Q24:

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

  • A 1 5 5 5 2 0
  • B 6 4 8 0
  • C 4 . 8 3 Γ— 1 0 8
  • D 3 . 0 3 Γ— 1 0 1 9
  • E 9 7 2 0

Q25:

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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