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In 2D space the point $(a,b)$ are said to be a lattice point if $a$, $b$ are integers. Two lattice points $(a,b)$ and $(c,d)$ are neighbours if they agree in one of the coordinates and differ in the other coordinate by $1$ . Every pair of neighbours is connected by a directed line with direction coinciding with the positive direction of the coordinate axis to which it is parallel. In how many ways one can move from $(0,0)$ to $(a,b)$ along the directed paths ?

Jessie
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Amogh Srivastava
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1 Answers1

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If we pay close attention to the situation, we have to make $a$ moves in the right direction, and $b$ moves in the upward direction.

We will find ourselves on $(a,b)$ after making those moves, no matter in what order we make those moves.

Now, this problem has switched to: In how many ways can we arrange $a$ R's and $b$ U's.

For instance, if we have to find the number of ways one can move from $(0,0)$ to $(4,5)$, We will have to arrange $\underbrace{RRRR}_\text{4 times} \overbrace{UUUUU}^\text{5 times}$ in all possible ways, i.e., $\frac{9!}{4!5!}$

Now, generalising it for $(a,b)$, the number of ways = $\frac{(a+b)!}{a! \times b!}$

letsintegreat
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