Chapter 10 Counting Methods

Chapter 10: Counting Methods 10.1 Counting by Systematic Listing 10.2 Using the Fundamental Counting Principle 10.3 Using Permutations and Combinations 10.4 Using Pascal’s Triangle 10.5 Counting Problems Involving “Not” and “Or”

Using Pascal’s Triangle Section 10-4 Using Pascal’s Triangle

Using Pascal’s Triangle Construct Pascal’s triangle and recognize the relationship between its entries and values of nCr. Use Pascal’s triangle to solve applications involving combinations.

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 and right are equally likely. In each circle the number of different routes that could bring us to that point are recorded.

Pascal’s Triangle START 1 1 1 1 2 1 1 1 3 3 1 4 1 4 6

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.

Pascal’s Triangle row and so on 1 2 3 4 6 5 10 15 20 7 21 35 1 2 3 4 5 1 2 3 4 5 6 7 and so on

Combination Values in Pascal’s Triangle The “triangle” possesses many properties. In counting applications, entry number r in row number n is equal to nCr – the number of combinations of n things taken r at a time. The next slide shows part of this correspondence.

Combination Values in Pascal’s Triangle row 1 2 3 4 5 0C0 1C0 1C1 2C0 2C1 2C2 3C0 3C1 3C2 3C3 4C0 4C1 4C2 4C3 4C4 5C0 5C1 5C2 5C3 5C4 5C5 and so on

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 7C5 from row 7 of Pascal’s triangle. The answer is 21.

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 and 3 men? Solution To select the women (2), we have 3C2 ways. To select the men (3), we have 4C3 ways. This gives a total of:

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 set. This answer is from row 6 of Pascal’s triangle:

Summary of Tossing Six Fair Coins Number of Heads n Ways to Have Exactly n Heads 1 2 3 4 5 6