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

Q1:

A class contains 27 boys and 29 girls. In how many ways can you select a team of 4 people from the class such that every member of the team is of the same sex?

Q2:

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 “1234” and “2134” 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?

Q3:

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.

Q4:

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.

Q5:

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

Q6:

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

Q7:

Let 𝑋 = { 𝑥 ∶ 𝑥 ∈ ℤ , 1 0 ≤ 𝑥 ≤ 1 6 } and 𝑌 = { { 𝑎 , 𝑏 } ∶ 𝑎 , 𝑏 ∈ 𝑋 , 𝑎 ≠ 𝑏 } . Determine the value of 𝑛 ( 𝑌 ) , where 𝑛 ( 𝑌 ) is the number of elements in 𝑌 .

Q8:

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?

Q9:

In how many diﬀerent ways can 71 people be picked from 74?

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:

If the elements of the set { 𝐴 , 𝐵 , 𝐶 , 𝐷 } are arranged on a circle, determine the number of polygons which can be made using these points as vertices.

Q13:

At a college, there were 120 ways to select 119 students to attend a seminar. Determine the number of students at the college.

Q14:

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

Q15:

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

Q16:

Given 102 points arranged in a circle, determine the number of line segments which can be formed using these points.

Q17:

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?

Q18:

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.

Q19:

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?

Q20:

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

Q21:

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

Q22:

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

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