1. SECTION 10-4 Using Pascal’s Triangle Slide 10-4-1

2. USING PASCAL’S TRIANGLE Pascal’s Triangle Applications Slide 10-4-2

3. PASCAL’S TRIANGLE Slide 10-4-3 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.

4. PASCAL’S TRIANGLE Slide 10-4-4 1 11 112 133 1 START 164 4 1

5. PASCAL’S TRIANGLE Slide 10-4-5 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.

6. PASCAL’S TRIANGLE Slide 10-4-6 1 11 121 1331 14641 1510 51 1615201561 172135 2171 and so on row 0 1 2 3 4 5 6 7

7. COMBINATION VALUES IN PASCAL’S TRIANGLE Slide 10-4-7 The “triangle” possesses many 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.

8. COMBINATION VALUES IN PASCAL’S TRIANGLE Slide 10-4-8 0C00C0 1C01C01C11C1 2C02C02C12C12C22C2 3C03C03C13C13C23C23C33C3 4C04C04C14C14C24C24C34C34C44C4 5C05C05C15C15C25C25C35C35C45C45C55C5 and so on 0 1 2 3 4 5 row

9. EXAMPLE: APPLYING PASCAL’S TRIANGLE TO COUNTING PEOPLE Slide 10-4-9 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

10. EXAMPLE: APPLYING PASCAL’S TRIANGLE TO COUNTING PEOPLE Slide 10-4-10 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

11. EXAMPLE: APPLYING PASCAL’S TRIANGLE TO COIN TOSSING Slide 10-4-11 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:

12. SUMMARY OF TOSSING SIX FAIR COINS Slide 10-4-12 Number of Heads n Ways to Have Exactly n Heads 0 1 2 3 4 5 6