Gisela lives in a subdivision where the streets are laid out in a grid pattern. Her school is eight blocks south and six 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 four blocks east of Gisela's home. Explain your reasoning.
The final answer is 735.
Thanks for all the help that is provided:)
She first must go any permutation of SSSEEEE ways to get to the friends house.
That results in 7!/(3!4!) or 35 ways.
That leaves you with 5 southern paths, and 2 eastern paths.
the remaining trip can be done in 7!/(5!2!) or 21 ways.
What is 35 x 21 ?
735
Ah, the classic grid conundrum! Let's put on our mathematical comedy helmets and solve this merry riddle.
Now, for Gisela to reach her school, she must first drop by her friend's house, because, well, friends gotta stick together, right? Her friend's house is 3 blocks south and 4 blocks east.
Since Gisela's school is 8 blocks south and 6 blocks east of her own house, we can think of this as a little adventure. Gisela needs to travel a total of 11 blocks south and 10 blocks east to reach her destination with a pit stop at her friend's house.
Now, let's assign some unique letters to each step of her journey. We'll use "S" for south and "E" for east. So, Gisela needs to take a total of 11 "S" steps and 10 "E" steps.
To find the number of different routes she can take, we can use a fun little formula called the binomial coefficient. We calculate it like this: (11+10)! / (11! * 10!).
Ready for some number clowning?
(11+10)! = 21! = 21 * 20 * 19 * ... * 1
11! = 11 * 10 * 9 * ... * 1
10! = 10 * 9 * ... * 1
Now, let the canceling extravaganza begin!
(11+10)! / (11! * 10!)
= (21 * 20 * 19 * ... * 1) / [(11 * 10 * 9 * ... * 1) * (10 * 9 * ... * 1)]
= (21 * 20 * 19 * ... * 12) / (10 * 9 * ... * 1)
And the final punchline...
(21 * 20 * 19 * ... * 12) / (10 * 9 * ... * 1)
= 735
So, Gisela has a whopping 735 different routes she can take to school with her friend's house stop included. Bravo, Gisela, you have choices!
To find the number of different direct routes Gisela can take when she bicycles to school, we can use the concept of combinatorics.
First, let's break down the problem into smaller steps:
1. Gisela's home to her friend's home:
Gisela needs to travel three blocks south and four blocks east to reach her friend's home. The number of different ways to do this can be calculated using the combination formula: C(3+4, 4) = C(7, 4) = 35. This means there are 35 different routes from Gisela's home to her friend's home.
2. Friend's home to school:
From her friend's home to school, Gisela needs to travel eight blocks south and six blocks east. Using the combination formula again: C(8+6, 6) = C(14, 6) = 3003. This means there are 3003 different routes from her friend's home to school.
3. Total routes:
To calculate the total number of routes, we need to multiply the number of routes for each step. So, the total number of routes will be 35 * 3003 = 105,105.
However, we need to consider that Gisela always stops by her friend's home. Therefore, all the routes that start from her friend's home and end at school will be counted twice, as they were already counted from Gisela's home to her friend's home.
To correct this, we subtract the number of routes that go directly from her friend's home to school without going through Gisela's home:
C(8+6, 6) = C(14, 6) = 3003.
Finally, the total number of different direct routes from Gisela's home to school, considering the stop at her friend's home, is 105,105 - 3003 = 102,102.
Therefore, the correct answer is 102,102, not 735 as mentioned.
To find the number of different direct routes Gisela can take when cycling to school, we can use the formula for combinations.
First, let's calculate the number of ways Gisela can travel from her home to her friend's home (3 blocks south and 4 blocks east). This can be calculated by using the formula for combinations:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of steps to be taken and k is the number of south steps or east steps.
In this case, Gisela takes 3 south steps and 4 east steps. Plugging these values into the formula:
C(7, 3) = 7! / (3!(7-3)!) = 7! / (3!4!)
= (7 * 6 * 5 * 4!) / (3! * 4!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35
So, Gisela can take 35 different routes from her home to her friend's home.
Next, let's calculate the number of ways Gisela can travel from her friend's home to her school (8 blocks south and 6 blocks east):
C(14, 8) = 14! / (8!(14-8)!)
= (14 * 13 * 12 * 11 * 10 * 9 * 8!) / (8! * (6 * 5 * 4 * 3 * 2 * 1))
= (14 * 13 * 12 * 11 * 10 * 9) / (6 * 5 * 4 * 3 * 2 * 1)
= 3003
So, Gisela can take 3003 different routes from her friend's home to her school.
To find the total number of different direct routes Gisela can take, we multiply the number of routes from her home to her friend's home by the number of routes from her friend's home to her school:
Total routes = 35 * 3003 = 105,105
Therefore, Gisela can take 105,105 different direct routes when cycling to school if she always stops by her friend's home.