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Josh starts at the point (0,0) and needs to travel to his home at the point (6,7). There is a fire at the point (2,3) so he cannot travel there. How many ways are there for Josh to reach his home if he can only move up or right?

 Dec 30, 2023
 #1
avatar+737 
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We can solve this problem by considering the number of moves right and up that Josh makes.

 

Josh needs to move 6 steps to the right and 7 steps up, for a total of 13 moves.

 

Each move can be either right or up, so there are 2 choices for each of the 13 moves.

 

Therefore, the total number of ways to reach his home is 2^13 = 8192.

 

However, this includes paths that go through the fire at (2,3). To exclude these paths, we need to consider how many paths go through the fire and subtract them from the total.

 

A path goes through the fire if it includes a move right at (1,2) and a move up at (2,2).

 

There are 11 ways to choose which of the remaining 11 moves will be the right move at (1,2).

 

For each of these 11 choices, there are 6 ways to choose which of the remaining 6 moves will be the up move at (2,2).

 

Therefore, there are 11 * 6 = 66 paths that go through the fire.

 

Finally, we subtract the number of paths that go through the fire from the total number of paths to find the number of paths that avoid the fire:

 

8192 total paths - 66 paths through the fire = 8126 paths that avoid the fire.

 

Therefore, there are 8126 ways for Josh to reach his home without going through the fire.

 Dec 31, 2023
 #2
avatar+126665 
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Total number  of ways to get from (0,0) to (6,7)  less the number of ways to get from (0,0) to ( 2,3)

 

C(13,6)  - C(5,2)  =   1716 -10  =  1706

 

cool cool cool 

 Dec 31, 2023

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