April 13, 2009 Counting methods (Part 4): More combinations page 1 Counting methods (Part 4): More combinations Recap of last lesson: The combination number n C r is the answer to this counting question: how many ways can r out of n objects be selected without an order? To calculate n C r, set up a fraction as follows: on the top, multiply r whole numbers counting downward from n; on the bottom, multiply r whole numbers counting upward from 1. In today s lesson you ll see: more counting problems that are answered by combination numbers a number pattern called Pascal s Triangle containing all of the combination numbers Problems: sequences of coin flips 1. A coin has two sides called heads (H) and tails (T). Suppose that a coin is flipped 3 times. Think of all of the possible ways that the sequence of 3 flips could turn out. a. Write all of the possible sequences of 3 coin flips. (example: HTH) b. For each of the coin flip sequences, determine which of the following categories it falls into, based on the number of heads and the number of tails. Finish filling in the chart below. 3 0 HHH 1 2 1 HHT, HTH, THH 3 1 2 0 3 2. Repeat the previous problem, but flipping 4 coins instead of 3. 4 0 3 1 2 2 1 3 0 4
April 13, 2009 Counting methods (Part 4): More combinations page 2 How coin flips are connected to combination numbers Here s what you should have found out about flipping 3 coins: 3 0 HHH 1 2 1 HHT, HTH, THH 3 1 2 TTH, THT, HTT 3 0 3 TTT 1 Look at the total numbers in the right column (1, 3, 3, 1). It turns out that each of them is actually a combination number involving a choice out of a total of 3: 3 C 0 = 1, 3 C 1 = 3, 3 C 2 = 3, and 3 C 3 = 1. This means that you could have figured out those numbers without writing out all the sequences of H s and T s, just by evaluating some C numbers. # of heads # of tails # of sequences of this kind 3 0 3C 0 = 1 2 1 3C 1 = 3 1 2 3C 2 = 3 0 3 3C 3 = 1 And here s what you should have found out about flipping 4 coins: 4 0 HHHH 4C 0 = 1 3 1 HHHT, HHTH, HTHH, THHH 4C 1 = 4 2 2 HHTT, HTHT, HTTH, THHT, THTH, TTHH 4C 2 = 6 1 3 TTTH, TTHT, THTT, HTTT 4C 3 = 4 0 4 TTTT 4C 4 = 1
April 13, 2009 Counting methods (Part 4): More combinations page 3 More coin flip problems 3. Fill in this chart about what happens when flipping a coin 6 times. Hint: The answers you need to evaluate 6 C 0, 6 C 1, etc. You can get these values from your calculator. # of heads # of tails # of sequences of this kind 6 0 5 1 4 2 3 3 2 4 1 5 0 6 4. Fill in this chart about what happens when flipping a coin 8 times. # of heads # of tails # of sequences of this kind 8 0 7 1 6 2 5 3 4 4 3 5 2 6 1 7 0 8 5. Suppose a coin is flipped 10 times. How many coin flip sequences are there that contain exactly 3 heads? 6. Suppose a coin is flipped 12 times. How many coin flip sequences are there that contain exactly 5 tails?
April 13, 2009 Counting methods (Part 4): More combinations page 4 Conclusion: repeated binomial experiments A binomial experiment is an action that has two possible outcomes (example: coin flip). A repeated binomial experiment is a two-outcome action done multiple times. What we ve observed involving coin flips actually applies to all binomial experiments. Here s the general conclusion: Suppose that a binomial experiment is repeated n times. Getting a certain outcome r-out-of-n times can occur in n C r different ways. Pascal s Triangle of combination numbers This triangular number pattern called Pascal s Triangle is a good way to find combination numbers without having to multiply or use a calculator. Each number in the pattern is the sum of the two numbers above it. For example, 5 C 2 = 10 comes from adding the 4 and 6 above it. 0C 0 1C 0 1 C 1 2C 0 2 C 1 2 C 2 3C 0 3 C 1 3 C 2 3 C 3 4C 0 4 C 1 4 C 2 4 C 3 4 C 4 5C 0 5 C 1 5 C 2 5 C 3 5 C 4 5 C 5 6C 0 6 C 1 6 C 2 6 C 3 6 C 4 6 C 5 6 C 6 = 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 So now we have three possible ways to get the value of a combination number n C r : by writing a fraction and evaluating it by using the calculator shortcut in the MATH PRB menu by writing the Pascal s Triangle pattern Problems 7. Write the next row of the triangles above, then identify the value of 7 C 3. 8. Write another row of the triangles above, then identify the value of 8 C 5.
April 13, 2009 Counting methods (Part 4): More combinations page 5 More combination number problems 9. A baseball team has played 7 games. Each game is either a win (W) or a loss (L). Think about all the possible sequences of wins and losses (example: W W L W L L W). Fill in the table below. Hint: Think of wins-and-losses just like heads-and-tails. # of wins # of losses # sequences of this kind 7 0 6 1 5 2 4 3 3 4 2 5 1 6 0 7 10. Suppose that an Italian Sub sandwich can be ordered with or without each of the following items, in any combination: lettuce, tomato, salt, pepper, oregano For each number of items listed below, how many different Italian Subs exist? Write your answers first in n C r form, then as plain numbers. 0 items 1 item 2 items 3 items 4 items 5 items
April 13, 2009 Counting methods (Part 4): More combinations page 6 11. To get the answers to the next problem you will need to make Pascal s Triangle down to the row that begins 1 10 Write out this number triangle in the space here. The beginning of it is shown. Try to figure out the rest without looking back at the previous pages. 1 1 1 1 2 1 12. A pizzeria offers 9 different pizza toppings. How many different pizzas are there with each of the following numbers of toppings? 0 toppings: 5 toppings: 1 topping: 6 toppings: 2 toppings: 7 toppings: 3 toppings: 8 toppings: 4 toppings: 9 toppings: 13. A competing pizzeria offers 10 different pizza toppings. How many different pizzas are there with each of the following numbers of toppings? 0 toppings: 5 toppings: 1 topping: 6 toppings: 2 toppings: 7 toppings: 3 toppings: 8 toppings: 4 toppings: 9 toppings: 10 toppings:
April 13, 2009 Counting methods (Part 4): More combinations page 7 14. From the numbers {1, 2, 3, 4, 5, 6, 7} you are asked to choose 3 of the numbers. How many different ways can these choices be made? 15. You are visiting a city, and you have tickets for a play at a theater that is 3 blocks south and 4 blocks east from your current location. Here is a map illustrating the situation. you theater You want to walk to the theater using one of the shortest possible routes. There are many different routes you could choose; here s a picture of one of them. you How many possible routes are there? theater Hint: Each route consists of walking along 7 city block segments: 3 going south and 4 going east. Think about when the choices are made to walk south. For example, the route shown above involves walking south as the 2nd, 4th, and 5th segments. This means that choosing a route is equivalent to picking 3 numbers from {1, 2, 3, 4, 5, 6, 7}; in this case the choices are {2, 4, 5}. How many different ways can those choices be made?