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With rectangular axes of coordinates, the number of paths from $(1,1)$ to $(8,10)$ via $(4,6)$, where each step from any point $(x,y)$ is either to $(x,y+1)$ or to $(x+1, y)$, is ____
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We can draw the rectangular axes of coordinates,

We have, each step path from any point $(x,y)$ is either to $(x, y+1)$ or to $(x+1, y)$ 

That mean, the path will either go along the $x$-axis or $y$-axis.

To reach from $(1,1)$ to $(4,6),$ we need,

  • $3$ steps of $x$
  • $5$ steps of $y$

$ \underbrace{x\;x\;x \; y\;y\;y\;y\;y}_{\qquad \text{In any order}\\ \text{(we can use permutation)}}$  (This is one possible path)

So, possible paths from $(1,1)$ to $(4,6) = \frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6 \times 5!}{ 3 \times 2 \times 1 \times 5!} = 56 \; \text{ways.}$

Similarly, from $(4,6)$ to $(8,10),$ we need 

  • $4$ steps of $x$
  • $4$ steps of $y$

$ \underbrace{x\;x\;x \;x \; y\;y\;y\;y\;y}_{\qquad \text{In any order}\\ \text{(we can use permutation)}}$  (This is one possible path)

So, possible paths from $(4,6)$ to $(8,10) = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5 \times 4!}{ 4 \times 3 \times 2 \times 1 \times 4!} = 70 \; \text{ways.}$

Now, the total number of paths from $(1,1)$ to $(8,10) = 56 \times 70 = 3920 \; \text{ways.}$

$\therefore$ The total number of paths from $(1,1)$ to $(8,10)$ via $(4,6)$ is $3920.$

Correct Answer $:3920$

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