Question
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?
Answers
GPT 3.5
We can use a grid to represent Josh's movements. The number of ways for Josh to travel from (0,0) to (6,7) can be represented by the number of paths from the top-left corner to the bottom-right corner of a 6 by 7 grid, moving only up or right.
To calculate this, we can use Pascal's Triangle. Each point on the grid represents a number, starting from 1 at the top-left corner, and each number represents the number of ways to reach that point from the top-left corner.
By using Pascal's Triangle, we can fill in the grid accordingly:
\[
\begin{array}{ccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
1 & \boxed{3} & 6 & 10 & 15 & 21 & 28 \\
1 & 4 & 10 & 20 & 35 & 56 & 84 \\
1 & 5 & 15 & 35 & 70 & 126 & 210 \\
1 & 6 & 21 & 56 & 126 & 252 & 462 \\
1 & 7 & 28 & 84 & 210 & 462 & 924 \\
\end{array}
\]
Therefore, there are $\boxed{28}$ ways for Josh to reach his home.
To calculate this, we can use Pascal's Triangle. Each point on the grid represents a number, starting from 1 at the top-left corner, and each number represents the number of ways to reach that point from the top-left corner.
By using Pascal's Triangle, we can fill in the grid accordingly:
\[
\begin{array}{ccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
1 & \boxed{3} & 6 & 10 & 15 & 21 & 28 \\
1 & 4 & 10 & 20 & 35 & 56 & 84 \\
1 & 5 & 15 & 35 & 70 & 126 & 210 \\
1 & 6 & 21 & 56 & 126 & 252 & 462 \\
1 & 7 & 28 & 84 & 210 & 462 & 924 \\
\end{array}
\]
Therefore, there are $\boxed{28}$ ways for Josh to reach his home.