Mathematics
Grade 6
15 min
Combinations
Combinations
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1
Introduction & Learning Objectives
Learning Objectives
Define what a combination is in mathematics.
Distinguish between situations where the order of selection matters and where it does not.
Systematically list all possible combinations for a small set of items.
Use tree diagrams or organized lists to find unique combinations.
Solve simple word problems involving combinations by identifying and eliminating duplicate arrangements.
Explain why certain arrangements are considered the same combination.
Imagine you're picking two ice cream flavors from a list of three. Does it matter if you pick chocolate then vanilla, or vanilla then chocolate? 🍦🤔
In this lesson, you'll learn about combinations, which are ways to choose items where the order of selection doesn't change the group. Understanding c...
2
Key Concepts & Vocabulary
TermDefinitionExample
CombinationA selection of items from a larger group where the order in which the items are chosen does not matter. The group itself is what counts.Choosing 2 fruits from an apple, a banana, and an orange. Picking an apple then a banana is the same combination as picking a banana then an apple.
OrderThe sequence or arrangement in which items are selected or placed. In combinations, order is not important.If you are choosing two students for a committee, 'Alice and Bob' is the same as 'Bob and Alice', so the order doesn't matter.
GroupA collection of items. In combinations, we are interested in unique groups, regardless of how they were formed.A group of 3 friends (Sarah, Tom, Lily) is one combination, no matter who was chosen first, second, or...
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Core Formulas
The 'Order Doesn't Matter' Principle
When forming a group or selecting items, if changing the sequence of the chosen items does not create a new or different group, then it is a combination problem.
This principle helps you identify if a problem is about combinations. For example, if you pick two colors for a painting, 'red and blue' is the same as 'blue and red'. The order doesn't change the pair of colors.
Systematic Listing Strategy for Combinations
To find all unique combinations, list all possible arrangements first, then carefully identify and remove any arrangements that are just the same group of items in a different order.
Start by listing all possible ways to pick items (even if order matters initially). Then, go through your...
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Challenging
If you are told there are exactly 3 possible combinations when choosing 2 items from a small set, how many items were in the original set?
A.2
B.4
C.5
D.3
Challenging
There are 4 students in a study group. They form 6 unique pairs to review topics. If a 5th student joins the group, how many additional pairs can be formed?
A.4
B.5
C.10
D.1
Challenging
A customer at a salad bar with many toppings says, 'Choosing carrots then tomatoes is a different salad than choosing tomatoes then carrots.' Why is this thinking incorrect for a problem about combinations of toppings?
A.Because carrots and tomatoes taste the same.
B.Because the salad bowl can only hold two items.
C.Because the final group of toppings in the bowl is the same regardless of the picking order.
D.Because you always add the heaviest topping first.
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