Downtown Mathville is laid out as a square grid of 9 North-South streets and 9 East-West
streets (See diagram). Your apartment is located at the Southwest corner of downtown
Mathville. (See point T.) Your math classroom is located in a building that is 6 blocks
East and five blocks North of your apartment. (See point B. The building is on the
corner.) You know that it is an 11 block walk to math class and that there is no shorter
path (no cutting across a block diagonally). Your curious roommate (we’ll call her
Curious Georgia) asks how many different paths (of length 11 blocks – you don’t want
backtrack or go out of your way) could you take to get from your apartment to the math
class. Solve Curious Georgia’s math problem and give a careful explanation as to why
your answer is correct.
B
T
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I (we) did not use any resources such as the web or books other than our textbook.
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72
To solve Curious Georgia's math problem, we need to find the number of different paths of length 11 blocks from your apartment to the math class. Let's start by understanding the layout of Downtown Mathville.
Downtown Mathville is a square grid of 9 North-South streets and 9 East-West streets. The Southwest corner of Downtown Mathville is your apartment, denoted by the point T. Your math classroom is located in a building that is 6 blocks East and 5 blocks North of your apartment, denoted by the point B.
Since there is no shorter path (no cutting across a block diagonally), we will have to find all the possible paths that follow the given conditions:
1. The path must be of length 11 blocks.
2. The path must start at T (your apartment) and end at B (your math classroom).
3. The path should not backtrack or go out of the way.
To count the number of different paths, we can use the concept of combinations. We need to make a total of 11 steps, out of which 6 must be towards the East and 5 must be towards the North. This can be represented as a sequence of E's (East) and N's (North):
EEEEEENNNNN
We can think of this sequence as arranging these letters in a specific order. To count the number of different arrangements, we can use the concept of permutations.
The total number of permutations is given by:
P = (total number of steps)! / (number of East steps)! * (number of North steps)!
In our case, P = 11! / 6! * 5!
Now, let's calculate the value of P:
P = (11 * 10 * 9 * 8 * 7 * 6!) / (6! * 5!)
The 6! terms will cancel out:
P = 11 * 10 * 9 * 8 * 7 / 5!
We can simplify this further:
P = 11 * 10 * 9 * 8 * 7 / (5 * 4 * 3 * 2 * 1)
P = 2312
Therefore, there are 2312 different paths (of length 11 blocks) that you could take to get from your apartment to the math class in Downtown Mathville.
Please note that the calculation assumes that you cannot move in any other direction except East or North. Also, this solution is based on the information provided in the problem statement and does not consider any additional constraints or obstacles that may be present in Downtown Mathville.