Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 11: Counting Methods 11.1 Counting by Systematic Listing 11.2 Using the Fundamental Counting Principle 11.3 Using Permutations and Combinations 11.4 Using Pascal’s Triangle 11.5 Counting Problems Involving “Not” and “Or”

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 11-4 Using Pascal’s Triangle

© 2008 Pearson Addison-Wesley. All rights reserved Using Pascal’s Triangle Pascal’s Triangle Applications

© 2008 Pearson Addison-Wesley. All rights reserved Pascal’s Triangle The triangular array on the next slide represents “random walks” that begin at START and proceed downward according to the following rule. At each circle (branch point), a coin is tossed. If it lands heads, we go downward to the left. If it lands tails, we go downward to the right. At each point, left an right are equally likely. In each circle the number of different routes that could bring us to that point are recorded.

© 2008 Pearson Addison-Wesley. All rights reserved Pascal’s Triangle START

© 2008 Pearson Addison-Wesley. All rights reserved Pascal’s Triangle Another way to generate the same pattern of numbers is to begin with 1s down both diagonals and then fill in the interior entries by adding the two numbers just above the given position. The pattern is shown on the next slide. This unending “triangular array of numbers is called Pascal’s triangle.

© 2008 Pearson Addison-Wesley. All rights reserved Pascal’s Triangle and so on row

© 2008 Pearson Addison-Wesley. All rights reserved Combination Values in Pascal’s Triangle The “triangle” possesses may properties. In counting applications, entry number r in row number n is equal to n C r – the number of combinations of n things taken r at a time. The next slide shows part of this correspondence.

© 2008 Pearson Addison-Wesley. All rights reserved Combination Values in Pascal’s Triangle 0C00C0 1C01C01C11C1 2C02C02C12C12C22C2 3C03C03C13C13C23C23C33C3 4C04C04C14C14C24C24C34C34C44C4 5C05C05C15C15C25C25C35C35C45C45C55C5 and so on row

© 2008 Pearson Addison-Wesley. All rights reserved Example: Applying Pascal’s Triangle to Counting People A group of seven people includes 3 women and 4 men. If five of these people are chosen at random, how many different samples of five people are possible? Solution Since this is really selecting 5 from a set of 7. We can read 7 C 5 from row 7 of Pascal’s triangle. The answer is 21

© 2008 Pearson Addison-Wesley. All rights reserved Example: Applying Pascal’s Triangle to Counting People Among the 21 possible samples of five people in the last example, how many of them would consist of exactly 2 women? Solution To select the women (2), we have 3 C 2 ways. To select the men (3), we have 4 C 3 ways. This gives a total of

© 2008 Pearson Addison-Wesley. All rights reserved Example: Applying Pascal’s Triangle to Coin Tossing If six fair coins are tossed, in how many different ways could exactly four heads be obtained? Solution There are various “ways” of obtaining exactly four heads because the four heads can occur on different subsets of coins. The answer is the number of size- four subsets of a size-six subset. This answer is from row 6 of Pascal’s triangle:

© 2008 Pearson Addison-Wesley. All rights reserved Summary of Tossing Six Fair Coins Number of Heads n Ways to Have Exactly n Heads