1. Chapter 10
Counting Methods

2. 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”

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

4. 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.

5. 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.

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

7. 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.

8. Pascal’s Triangle row and so on 1 2 3 4 6 5 10 15 20 7 21 35 1 2 3 4 51 2 3 4 5 6 7
and so on

9. 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.

10. 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

11. 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.

12. 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:

13. 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:

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