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- 1. Probability: Study Time Situation: The final provincial math exam is coming up and Christine and 4 of her friends have decided to create a study group. They meet up at John's house everyday. Since Christine is the one that lives the farthest from John, she decides that she will pick up her other three friends on the way.
- 2. This is a layout of how to get to everyone's houses. Christine's home Lucas Store Kelly Shelly John
- 3. First. • If Christine could only travel South and East, how many ways can she get to Lucas? Christine's home Lucas Store Kelly Shelly John
- 4. Answer.. There is SIX different ways to get to Lucas.
- 5. How I got this answer. Well the first thing you have to know is where both of the points are. Point A Point A would be Christine's house Point B would be Lucas' house. Point B The first thing you have to find out is 'how many ways can you get to the first corner'. A 1 Well as you can see in this diagram, the outer corner of the street is labelled as 1. This is because there is only one way for you to to go in each direction, 1 South or East. A 1 To get the resultant for the next point, what you would have to do is add the + two numbers together to get the answer for the bottom corner of the box. 1 2 This technique was discovered by Pascal's Triangle.
- 6. Pascal's Triangle Let me tell you a little more of what it is. Pascal's triangle is a arrangement of numbers (binomial numbers) that is formed into a triangle. *Hence the name* There are many patterns that you can see on Pascal's triangle. These are just some examples of them. 0 Example 1 2 All the numbers appear in 2 1 Example 3 order, diagonally. Fibonacci numbers. This 22 is the one that we will be using. Each term is the sum of the previous terms. 1,1,2,3,5,8,13,21 ... Example 2 All the sums of the rows in the triangle are equal to the power of 2
- 7. This is how it would look when you apply Pascal's Triangle into the problem. ( The Fibonacci numbers ) Christine's home 1 1 1 1 1 1 2 3 4 5 6 1 3 6 10 15 21 Lucas 1 4 10 20 35 56 15 1 5 Store 35 70 126 Kelly 1 6 21 56 126 252 1 7 28 84 462 210 1 1 1 Shelly 1 2 3 4 From here, you can see the maximum amount of ways to get to each 1 3 6 10 1 1 person's home. 2 3 1 John
- 8. Christine's home 1 1 1 1 1 Now how many 1 4 5 6 possible ways can you get from Christine's 2 3 1 3 Lucas 6 10 15 21 house the John's 1 4 10 20 35 56 house? 15 1 5 Store 35 70 126 Kelly 1 6 21 56 126 252 7 28 84 462 1 210 1 1 1 Shelly Using the Fibonacci numbers you can 1 2 3 4 determine how many ways you can get there now. 1 3 6 10 1 1 All you have you have to do is take the sum 2 3 1 of every corner and you multiply them to get John to John's house.
- 9. So you would do 462 x 10 x 3 = 12,780 ways to get to John's house. For Christine to go to John's house, she would get 12,780 different possible way's that she can get there.

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