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  4. Note the arrangement (1, 1), (1, 2), (1, 3), (2, 3), (3, 3), (3, 4), (4, 4). Here we start from (1, 1), then increase one of the co-ordinates by 1 and repeat the same until we reach (4, 4). For example (1, 1), (2, 1), (2, 2), (2, 3), (3.3), (3, 4), (4, 4) is another such arrangement. The number of such arrangements is

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Q. Note the arrangement $ (1, 1), (1, 2), (1, 3), (2, 3), (3, 3), (3, 4), (4, 4)$. Here we start from $(1, 1)$, then increase one of the co-ordinates by $1$ and repeat the same until we reach $ (4, 4)$. For example $ (1, 1), (2, 1), (2, 2), (2, 3), (3.3), (3, 4), (4, 4)$ is another such arrangement. The number of such arrangements is

Permutations and Combinations

Solution:

There are six steps in all, three of one kind (increasing $x$ co-ordinate by $1$) and three of the other (increasing $y$ co-ordinate by $1$). So we have to arrange six steps, three of them alike and of one kind and the other three alike of the second kind.
$\therefore $ total no. of ways $=\frac{6\,!}{3\,! \,3\,!}$