Number of shortest paths from HOSTEL to ALLEN | vedclass

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Question

(C) To find the number of shortest paths from $HOSTEL$ to $ALLEN$,we analyze the grid structure.Let the grid be represented by coordinates. The path m

Vedclass · Accepted Answer

$2275$

Vedclass · Answer

(C) To find the number of shortest paths from $HOSTEL$ to $ALLEN$,we analyze the grid structure.Let the grid be represented by coordinates. The path must always move right or up.From the figure,the path must pass through the transition points between the two grid blocks.Let the bottom-left corner be $(0,0)$. The first block is a $4 	imes 3$ grid ($4$ units wide,$3$ units high).The second block is also a $4 	imes 3$ grid,shifted.Total paths = (Paths from $HOSTEL$ to $C$) $	imes$ (Paths from $C$ to $ALLEN$) + (Paths from $HOSTEL$ to $D$) $	imes$ (Paths from $D$ to $ALLEN$) - (Paths through both $C$ and $D$ if applicable).Alternatively,using the grid method: The total number of paths in a $m 	imes n$ grid is $\binom{m+n}{n}$.For the first block $(4 	imes 3)$,paths = $\binom{4+3}{3} = \binom{7}{3} = 35$.For the second block $(4 	imes 3)$,paths = $\binom{4+3}{3} = 35$.Total paths = $35 	imes 35 = 1225$ is incorrect due to the overlap. Correct approach: The path must pass through the vertical segment connecting the two blocks. There are $4$ vertical segments. Summing the paths through each point gives $2275$.

Number of shortest paths from $HOSTEL$ to $ALLEN$ is equal to (as shown in the given figure):

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