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Geometry

Using the combination formula, find the number of subsets of a set according to the given conditions.

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

- Using the combination formula, find the number of subsets of a set which contain a certain number of elements.
- Using the combination formula, find the number of subsets of a set which contain either of two given certain numbers of elements.
- Using the combination formula, find the number of subsets of a set which contain a particular element.
- Using the combination formula, find the number of subsets of a set which contain at most a certain number of elements.

Combinations and using the Combination Formula.

A combination is an arrangement of objects in which order doesn't matter.

For example,

Suppose there are 3 books on a shelf - a history book (H), a math book (M) and a science book (S). If Andre wants to take any 2 books out, the possibilities are:

Since the possibilities in the second column are the same as the ones in the first column, we won't count them. So we are left with three possibilities (HM, HS and MS) which are combinations of books that Andre can select.

In order to calculate the number of combinations without writing each possibility out, we can use the Combination Formula to find the answer quickly.

The Combination Formula can be used to find all of the different ways to arrange r items out of n items.

The n represents the total number of items and the r represents how many things you are choosing.So, returning to our book example, there are 3 books and Andre would like to take out 2 of them.

The combination is also used when calculating the subsets of a set. In the Activity Object, there are real-life problems related to the subsets of a set which are thus solved using the combination formula. The learners are given the number of programs offered at a local gym and requirements for each flyer template and then asked to find the number of possible flyer templates needed.

Calculating the number of subsets of a set which contain a certain number of elements

Sample Problem: How many flyer templates are needed if there are 5 different types of programs and each flyer includes 3 of them?

Note that this problem is equal to the following problem: Find the number of subsets with 3 elements for set with 5 elements.

This example is actually a simple combination problem, since the order is not important. Therefore, we can use the Combination Formula:

Calculating the number of subsets of a set which contain a particular element.

Sample Problem: How many flyer templates are needed if there are 8 different types of programs and each flyer includes 4 types, one of which must be tennis?

Note that this problem is equal to the following problem: Find the number of subsets with 2 elements for the set with 8 elements when one of them is tennis.

Since one of the 8 elements is fixed, we need to choose the remaining 3 types of programs from the remaining total number of programs, which is 7.

Calculating the number of subsets of a set which contain either of two given certain numbers of elements.

Sample Problem: How many flyer templates are needed if there are 5 types of programs and each flyer includes 3 or 4 of them?

Note that this problem is equal to the following problem: Find the number of subsets with 3 or 4 elements for the set with 5 elements.

In this case, we need to add the results of two cases using the principle of counting by addition:

Calculating the number of subsets of a set which contain at most a certain number of elements.

Sample Problem: How many flyer templates are needed if there are 4 types of programs and each flyer includes at most 3 different types of programs?

Note that this problem is equal to the following problem: Find the number of subsets with at most 3 elements for the set with 4 elements.

We need to consider subsets having 0, 1, 2, or 3 elements.

Method 1:

First let's look at the four possible elements:

=Number of flyer templates which include no information about previous programs

=Number of flyer templates for members who took part in one type of program

=Number of flyer templates for members who took part in two different types of programs

=Number of flyer templates for members who took part in three different types of programs

Then we would need to use the principle of counting by addition to find the answer.

Method 2:

We can find the answer an easier way by using the formula , which is for finding all subsets of a set with n elements. It also equals the sum of the number of subsets with a certain number of elements.

Since we have 4 elements, gives all the subsets of a set, which is equal to the sum of all the

If we move to the left side of the equation, we get what the question is asking for.

Using the Combination Formula, we can solve:

Notice that this is equal to:

The Number of Proper Subsets = , where n is the number of elements.

Approximate Time | 20 Minutes |

Pre-requisite Concepts | combinations, combination formula, principle of counting by addition, sets, subsets |

Course | Geometry |

Type of Tutorial | Skills Application |

Key Vocabulary | combination formula, element, probability |