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# ZingPath: Counting Principles

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## Counting Principles

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### Lesson Focus

#### Counting Problems: Number of Parallelograms

Geometry

Find the number of parallelograms by finding the number of combinations and applying the principle of counting by multiplication.

### Now You Know

After completing this tutorial, you will be able to complete the following:

• Use the combination formula to find the number of combinations of r objects from among n objects when order is not important.
• Apply the principle of counting by multiplication to specified problems involving independent events.

### Everything You'll Have Covered

In this Activity Object, the learner is given a word problem involving finding the number of parallelograms formed when m parallel lines intersect with other n transversal parallel vertical lines. In order to solve this problem, the combination formula and the principle of counting by multiplication will be used.

Combination Formula When the order is not important, you can find the number of selections of r objects from a set of n objects by using the combination formula:

For example, Eleven students put their names in a hat to pick three names for a committee. How many different ways can the three names be selected out of the hat?

Principle of Counting by Multiplication

There are several counting methods, but this Activity Object focuses on the principle of counting by multiplication:

If an event occurs in m ways and another event occurs independently in n ways, then the two events can occur in m × n ways.

For example,

If there are shirts for sale in 3 colors (red, blue, and yellow) and come in 4 sizes (small, medium, large, and X-large), how many different shirts are available?

3 × 4 = 12 different shirts

The following key vocabulary terms will be used throughout this Activity Object:

• combination - an arrangement of objects in which order does not matter
• counting principle - states that if an event has m possible outcomes and another independent event has n possible outcomes, then there would be m × n possible outcomes for the two events together
• horizontal - at a right angle to a vertical line and parallel to level ground
• parallel lines - distinct lines lying in the same plane and never intersect each other
• parallelogram - a quadrilateral with two pairs of parallel and congruent sides
• point of intersection - the point at which two or more lines intersect
• transversal - a line that intersects two or more other lines at different points
• vertical - the direction of standing or pointing up

### Tutorial Details

 Approximate Time 20 Minutes Pre-requisite Concepts principle of counting by multiplication, combination formula Course Geometry Type of Tutorial Problem Solving & Reasoning Key Vocabulary combination, counting principle, parallelograms