Four letters, two `'a'` and two `'b'` are filled into `16` cells of a matrix as given. It is required that each cell contains atmost one letter and each row or column cannot contain same letters. Then the number of ways the matrix can be filled is

(A) `3600` (B) `5200` (C) `3960` (D) `4120`

Options

A. `3600`
B. `5200`
C. `3960`
D. `4120`

Answer

C

Explanation

`(c )` Selecting two rows for two `'a'` in `"^(4)C_(2)` ways. Now first `'a'` can be put in any row in `4` ways, Second `'a'` can be put in other row in `3` ways as no two `a'` are in same column.
Thus two `'a'` can be filled in
`"^(4)C_(2).4xx3=72` ways.
Similarly two `'b'` can be filled in `72` ways.
Thus two `'a'` and two `'b'` can be filled in `72^(2)` ways.
Now, we have no exclude the cases when
`(i)` both `'a'` and `'b'` are in same cell i.e., `72` ways (as after fixing `'a'` in `72` ways, `'b'` can be placed in this position)
`(ii)` one `'a'` and `'b'` are in same row or coloumn i.e., `72xx16` ways
Required number of ways `=72^(2)-72-72xx16=3960`