Permutations Question with Grid

[gracek]

Hi, I was wondering if I could get some help with this question. I understand the basic concepts but I can't quite grasp how to account for the blocked out squares. If anyone could explain this it would be much appreciated. Thank you!

"Determine the number of possible routes from Adam's house to Matt's house if you can only travel south and/or east. Hint: you can only travel when there is a line connecting."

 

[mmm4444bot]

Hello. How would you count the paths if there were no blocked-out squares?

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[imath]\;[/imath]

 

[gracek]

Hello. How would you count the paths if there were no blocked-out squares?

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[imath]\;[/imath]

There are 6 squares east and 5 squares south. I added them together to find 11. Then I did 11!/(5!6!)=462

I'm not sure if this is correct though.

 

[pka]

There are 6 squares east and 5 squares south. I added them together to find 11. Then I did 11!/(5!6!)=462
I'm not sure if this is correct though.

That is a very good start. But it does not answer the question.
The [imath]462[/imath] counts a number of paths in which there is a "south block" in one of the gray areas.
For example: [imath]SEEEE{\bf\color{red}S}SSSEE[/imath]. We cannot pass through a gray area.
How many of the [imath]462[/imath] paths have a "south block" in one of gray areas?
[imath][/imath][imath][/imath][imath][/imath]

 

[mmm4444bot]

462 counts a number of paths in which the is a "south block"

What does "in which the is" mean?

?

[imath]\;[/imath]

 

[gracek]

That is a very good start. But it does not answer the question.
The [imath]462[/imath] counts a number of paths in which there is a "south block" in one of the gray areas.
For example: [imath]SEEEE{\bf\color{red}S}SSSEE[/imath]. We cannot pass through a gray area.
How many of the [imath]462[/imath] paths have a "south block" in one of gray areas?
[imath][/imath][imath][/imath][imath][/imath]

Would you mind explaining how to determine this, please? Unfortunately I'm not really sure where to begin. I can't really explain what I'm having trouble with because I don't know how to approach the question in general. I think I just need a bit of direction in where to start. I've never taken a data management class before and this is my first experience with permutations so it would be much appreciated.

 

[Dr.Peterson]

Would you mind explaining how to determine this, please? Unfortunately I'm not really sure where to begin. I can't really explain what I'm having trouble with because I don't know how to approach the question in general. I think I just need a bit of direction in where to start. I've never taken a data management class before and this is my first experience with permutations so it would be much appreciated.

What he's saying is that you have a good start in counting all paths if the missing lines were present:

​

What you need to do is to subtract from that all paths that use each of these; fortunately, they are placed to that no path can go through more than one. So you need to count paths like this, which is not allowed:

​

You can use the same method you used for the total count, to count ways to get from A to the top red dot, and from the bottom dot to M. Do the same for each of the three red segments.

There are also other ways to solve this.

 

[pka]

Would you mind explaining how to determine this, please? Unfortunately I'm not really sure where to begin. I can't really explain what I'm having trouble with because I don't know how to approach the question in general. I think I just need a bit of direction in where to start. I've never taken a data management class before and this is my first experience with permutations so it would be much appreciated.

I am unable to copy & post the [imath]5\times 6[/imath] grid. Thank you Prof. Peterson!
There are five ways to leave house A and get to the first red point, Go down the corresponding red block.
From there there are [imath]\dfrac{5!}{2!\cdot3!}=10[/imath] ways to proceed.
So how many forbidden paths through the first red point?
Now there at two more forbidden blocks. Count the paths through each.

[imath][/imath][imath][/imath][imath][/imath]

 

[gracek]

I am unable to copy & post the [imath]5\times 6[/imath] grid. Thank you Prof. Peterson!
There are five ways to leave house A and get to the first red point, Go down the corresponding red block.
From there there are [imath]\dfrac{5!}{2!\cdot3!}=10[/imath] ways to proceed.
So how many forbidden paths through the first red point?
Now there at two more forbidden blocks. Count the paths through each.

[imath][/imath][imath][/imath][imath][/imat [/QUOTE][/imath]

What he's saying is that you have a good start in counting all paths if the missing lines were present:

View attachment 33461​

What you need to do is to subtract from that all paths that use each of these; fortunately, they are placed to that no path can go through more than one. So you need to count paths like this, which is not allowed:

View attachment 33464​

You can use the same method you used for the total count, to count ways to get from A to the top red dot, and from the bottom dot to M. Do the same for each of the three red segments.

There are also other ways to solve this.

Thank you for your help!

 

[gracek]

I am unable to copy & post the [imath]5\times 6[/imath] grid. Thank you Prof. Peterson!
There are five ways to leave house A and get to the first red point, Go down the corresponding red block.
From there there are [imath]\dfrac{5!}{2!\cdot3!}=10[/imath] ways to proceed.
So how many forbidden paths through the first red point?
Now there at two more forbidden blocks. Count the paths through each.

[imath][/imath][imath][/imath][imath][/imath]

Thank you for your responses, they were very helpful! I think I understand better now.

 

[Cubist]

Thank you for your responses, they were very helpful! I think I understand better now.

That's great! However, I do recommend that you try to respond to the question...

So how many forbidden paths through the first red point?

How many forbidden paths through the first red line, the one with the spots at both ends

This isn't to put you on the spot - deliberate pun - this is purely to help. We want to make sure that you know how to combine the number of ways to and from in order to find the actual number of ways through that forbidden red line