## 12.11.1A Permutations

Imagine you are rearranging 6 books on a shelf.   There are many different ways to arrange those books.  The fundamental counting principle can be used to determine how many books can be placed in each position.  As each book is used, there are fewer books to put into the next slot.  This ordered arrangement is also called a Permutation and there are different ways to figure out how many permutations for a given group are available.

Resources

• Notes

Assignment

• p524 #1-8, 15, 22, 23, C1 C3

Things you should be able to do after today:

• Understand and solve problems involving factorial notation
• determine the number of permutations of distinct (different) objects when all of the objects are used or only some of the objects are used

## 12.11.1B Permutations with Repetitions and Constraints

Sometimes the objects in a group cannot be distinguised.  For example, think of a group of scrabble tiles: "ALBAASR".  An ordered arrangement of these 7 letters will have fewer combinations than 7 different scrabble tiles, because we cannot distinguish the difference between the different A's.  We could have a different situation where we need to put certain items together, or keep certain items apart.  These introduce further constraints.  In both of these cases, we need to find out the total number of permutations, which will be different than if all of the symbols are unique

Resources:

• Notes

Assignment:

• p524 #9-11, 14, 16-21, 25 *29, 32

Things you should be able to do after today:

• determine the number of permutations in a group if all the letters are used and there are repetitions within the group
• determine the number of permutations in a group if all of the members are used and there are constraints on the positions of those symbols
• apply the fundamental counting principle to problems that involve two different groups

## 12.11.2 Combinations

Imagine getting dealt a hand of cards.  When you receive the cards, you rearrange them by rank and/or by suit.  Does it matter what order the cards come in?

When order does not matter, we call this a combination (as opposed to permutations).

Resources

• Notes

Assignment:

• p534 #1-11, 15, 17-21, *12 *16 *22 *23

## 11.3 Binomial Theorem

The expansion for powers of a binomial involves using either the numbers in Pascal's Triangle, or combinatorics. Today, you will explore the use of both in helping expand a polynomial that can be written as the powers of a binomial:

Resources

• Notes

Assignment

• p542 #1-6, 8-11, 13, 14
• p542 #7, 12, 15-21