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Part 1
In a certain​ city, there are six streets going​ north-south and five streets going​ east-west. How many street paths start at the southwest corner of the​ city, end at the northeast corner of the​ city, and have the shortest possible​ length?
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Part 1
There are 
  
enter your response here such paths.

To solve this problem, we need to determine the number of unique paths from the southwest corner to the northeast corner of the grid, following the shortest possible route.

### Step 1: Understanding the grid
Since there are 6 north-south streets and 5 east-west streets, we can think of the city grid as a \(6 \times 5\) grid. To reach the northeast corner from the southwest corner by the shortest route, we must travel 5 blocks east and 6 blocks north.

### Step 2: Total movements needed
For the shortest path, you must make:
- 5 eastward movements (E)
- 6 northward movements (N)

Thus, each path can be thought of as a sequence of 11 moves (5 E's and 6 N's).

### Step 3: Calculating unique paths
The problem now reduces to finding the number of unique sequences of these 11 moves, where we choose 5 positions out of 11 for eastward movements (the remaining 6 will be for northward movements). This is a combinatorial problem, where we calculate the binomial coefficient \( \binom{11}{5} \), which represents choosing 5 movements out of 11.

The formula for the binomial coefficient is:
\[
\binom{11}{5} = \frac{11!}{5!(11 - 5)!} = \frac{11!}{5! \times 6!}
\]

### Step 4: Computing the result
1. Calculate \(11! = 39,916,800\)
2. Calculate \(5! = 120\)
3. Calculate \(6! = 720\)

Now substitute:
\[
\binom{11}{5} = \frac{39,916,800}{120 \times 720} = \frac{39,916,800}{86,400} = 462
\]

### Answer:
There are **462** unique shortest paths from the southwest to the northeast corner.

Would you like further explanation, or have any questions?

Here are 5 related questions for further practice:
1. What if there were 7 north-south streets and 5 east-west streets? How many shortest paths would there be?
2. How many shortest paths would exist if the grid was a \(4 \times 4\) grid?
3. What if you were required to go through a particular intersection along the way? How would you calculate paths then?
4. How does the answer change if there are obstacles on certain intersections?
5. How would the solution change if one could make diagonal moves in addition to north-south and east-west?

**Tip:** Binomial coefficients are widely used in combinatorics and probability to count combinations, where the order of selection does not matter.

In a certain city, there are six streets going north-south and five streets going east-west. How many street paths start at the southwest corner of the city, end at the northeast corner of the city, and have the shortest possible length?

Learn how to calculate the number of shortest paths in a city grid with 6 north-south streets and 5 east-west streets. This problem involves combinatorics and binomial coefficients, and is suitable for students in Grades 9-12. The solution provides a step-by-step breakdown of how to apply the binomial coefficient formula to find the total number of paths.

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