Nancy lives in a subdivision where the streets are laid out in a grid pattern. Her school is six blocks south and four blocks east of her home. How many different direct routes could she take when she bicycles to school if she always stops by her friend’s home, which is three blocks south and three blocks

east of Nancy’s home.

I would do this with a sketch.

To find the number of different direct routes Nancy could take when she bicycles to school, we can use the concept of combinations.

First, let's visualize the grid pattern with Nancy's home at the starting point, her friend's home on the way, and the school as the final destination:

Nancy's home * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
* * * * * * * * * * * *
Friend's home * * * * * *
School * * * *

From Nancy's home to her friend's home, she needs to go 3 blocks south and 3 blocks east. From her friend's home to the school, she needs to go 3 blocks south and 4 blocks east.

To find the number of different direct routes, we need to find the number of combinations of choosing the number of times Nancy goes south and east, respectively, while considering the total number of blocks she needs to travel.

Since Nancy needs to go 3 blocks south and 3 blocks east before reaching her friend's home, we can arrange these movements in any order. This can be calculated as the number of combinations of choosing 3 movements out of a total of 6 movements. We can use the formula for combinations, which is:

nCr = n! / (r!(n-r)!)

n is the total number of movements, and r is the number of movements in a specific direction. In Nancy's case, n = 6 and r = 3.

nCr = 6! / (3!(6-3)!) = 6! / (3!3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20

So, there are 20 different routes from Nancy's home to her friend's home.

Similarly, from her friend's home to the school, she needs to go 3 blocks south and 4 blocks east. Using the same formula, nCr = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35.

Therefore, there are 35 different routes from her friend's home to the school.

To find the total number of different direct routes, we multiply the number of routes from Nancy's home to her friend's home (20) and from her friend's home to the school (35).

Total routes = 20 * 35 = 700

Hence, Nancy has 700 different direct routes she can take when she bicycles to school while stopping by her friend's home.