Combinatorics

Question

How many ways we can move from a point (a,b) to a point (i,j) in a coordinate system. Here i>a and j>b and at each step you can move to the right or in the upward direction.

Comments

Sorry u r right.

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i've added the answer for clear explanation.

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THANKS

Answer 1

We have to move from point $(a,b)$ to a point $(i,j) $

Hence there are $(i-a)$ right moves and $(j-b)$ up moves.  

All the moves can be arranged in any order and they will lead to point $(i,j)$

$(up,right,......,up,right)$

Total number of moves = $(i-a)+(j-b)$
Total number of right moves  =$(i-a)$
Total number of up moves = $(j-b)$

Total number of arrangements (permutations with repetition)  = $\Large \frac{\left ( (i - a) + (j - b) \right )!}{\underbrace{(i - a)!}_{\text{right moves}}\thinspace{\underbrace{(j - b!)}_{\text{up moves}}} }$

Comments

Well you are right, but you can also elaborate in terms of permuatation are there are (i-a) up moves repeated and (j-b) down moves repeated.

so total no. of permutation = ((i-a)+(j-b))! / (no. of repetition