Combinations

Combinations refer to the different ways in which a set of items can be arranged or selected without considering the order. In other words, combinations are used to find the number of ways to select a specific number of items from a larger set, where the order of selection does not matter.

Formula for Combinations

The number of combinations of n items taken r at a time is denoted by C(n, r) or nCr and is calculated using the formula:

C(n, r) = n! / (r! * (n - r)!)

Where n is the total number of items, r is the number of items to be selected, and ! represents the factorial of a number (the product of all positive integers up to that number).

Example

For example, if you have 5 different colored balls (red, blue, green, yellow, and orange) and you want to select 3 of them, the number of different combinations of 3 balls that can be selected from the 5 is given by:

C(5, 3) = 5! / (3! * (5 - 3)!) = 5! / (3! * 2!) = (5 * 4 * 3) / (3 * 2) = 10

So, there are 10 different combinations of 3 balls that can be selected from the 5.

Study Guide

Here are some key points to remember when working with combinations:

Combinations are used to find the number of ways to select items from a set without considering the order.
The formula for combinations is C(n, r) = n! / (r! * (n - r)!), where n is the total number of items, and r is the number of items to be selected.
Factorial (!) of a number is the product of all positive integers up to that number.
When using the combination formula, make sure to calculate the factorials accurately and simplify the expression.
Practice problems involving combinations to gain a better understanding of the concept.

Understanding combinations is important in various fields such as probability, statistics, and combinatorics. Mastery of this concept will enable you to solve a wide range of problems involving the selection of items from a set.

◂Math Worksheets and Study Guides Fifth Grade. Volume and Capacity

The resources above cover the following skills:

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes.

Use visualization, spatial reasoning, and geometric modeling to solve problems.

Use geometric models to solve problems in other areas of mathematics, such as number and measurement.

Measurement (NCTM)

Understand measurable attributes of objects and the units, systems, and processes of measurement.

Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute.

Carry out simple unit conversions, such as from centimeters to meters, within a system of measurement.

Apply appropriate techniques, tools, and formulas to determine measurements.

Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles.

Grade 5 Curriculum Focal Points (NCTM)

Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area

Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems.

Connections to the Grade 5 Focal Points (NCTM)

Measurement: Students' experiences connect their work with solids and volume to their earlier work with capacity and weight or mass. They solve problems that require attention to both approximation and precision of measurement.