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If there is a 4 X 4 grid and I have 4 letters (2 A,2 B).

In how many ways can I place them in the grid so that no two A's or no two B's are in the same horizontal row/vertical row.

A,B can be in the same row but 2 A's or 2B's or both cant be in the same row.

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1 Answer 1

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Placing $A$'s first, there are $(16 \times 9) / 2 = 72$ possibilities.

Next, try to place $B$'s. Without loss of generality, we can assume that the two $A$'s are at location $(1, 1)$ and $(2, 2)$.

I'll split the location of the first $B$ into 3 cases:

  1. $\{(2, 1), (1, 2)\}$. In this case, the second $B$ has $9$ locations available.
  2. $\{(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 2), (4, 1), (4, 2)\}$. In this case, the second $B$ has $8$ locations available.
  3. $\{(3, 3), (3, 4), (4, 3), (4, 4)\}$. In this case, the second $B$ has $7$ locations available.

Therefore, the answer is $$ 72 \times \left[ (2 \times 9) + (8 \times 8) + (4 \times 7) \right]/2 = 3960. $$

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