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There's a 6 by 6 grid and you're asked to start on the top left corner. Now your aim is to get to the bottom right corner. You are only allowed to move either right or down. You must never move diagonally or backwards.

I figured out the way to do this through writing number of ways to reach each cell in a box and got the answer to be 252

However I saw a formula for questions of this type which gives answer as 12!/(6!*6!)

I would like to know why the above formula doesn't work in this case and preferably where it's applicable

Thanks

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Your result is fine. Presumably you were writing numbers in the squares, of which there are five on a side. The question was following the lines of the grid, of which there are six on a side. You take $10$ moves to get from one corner to the other, of which you have to choose $5$ to be downwards, and $\frac {10!}{5!5!}=252$. They take $12$ moves to get from one corner to the other, have to choose $6$ to be down, and have $\frac {12!}{6!6!}=924$ choices. If you add one more row of squares on the top and right and continue your approach, you should find $924$ as well.

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If your grid is such that you need to make $6$ left and $6$ down movements, then what you are looking at is in how many ways can you arrange the letters "LLLLLLDDDDDD". This is the same as asking how many ways can we place $6$ L's in $12$ slots. This is exactly what is counted by the binomial coefficient $\binom{12}{6}=924$.

However, if your grid is such that you need to make $5$ left and $5$ down movements, then the same argument gives $\binom{10}{5}=252$.

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