Lesson Worksheet: Counting Using Combinations Mathematics • 12th Grade

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


Let 𝑋={𝑥𝑥,10𝑥16} and 𝑌={{𝑎,𝑏}𝑎,𝑏𝑋,𝑎𝑏}. Determine the value of 𝑛(𝑌), where 𝑛(𝑌) is the number of elements in 𝑌.


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


A school’s administration is choosing the colors of its new logo. The logo is made of three letters: CHS. They would like to have a different color for each letter. The options for the colors are red, green, blue, yellow, orange, and purple.

How many different ways can they color their logo?

The school administration decided that instead of having a different color for each letter, they would buy three out of the six colors, mix them all together, and then paint the whole logo in the resulting color.

How many color options do they now have for their logo?


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?


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.


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


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


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


Mia’s teacher divided the class into groups of 6 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 company’s security officers are ordering a new lock for the company’s front door. They are concerned about security, so they would like a lock with at least 100,000 different possible codes. They bought a lock whose code is formed of six distinct digits chosen from 09, believing that the order in which the digits are entered matters.

How many distinct possible door codes would such a lock have?

Unfortunately, when it arrived, they discovered that it was a genuine combination lock—that is, one for which order does not matter. How many distinct possible combinations does the lock have?

This lesson includes 36 additional questions and 323 additional question variations for subscribers.

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