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Find the number of ways in which A A A B...

Find the number of ways in which A A A B B B can be places in the square of fig as shown, so that no row remains empty. fig

Text Solution

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`2^(nd) " and" 4^(th)` row block has to be selected.
One row out of 1st, 3rd and 5th will contain two letters and remaining one letter each.
Selection of one row that contains two letters is done in `.^(3)C_(1)` ways.
Now two blocks can be selected from this row in `.^(3)C_(2)` ways.
From each of reamaining two row, one blocks can be selected in `.^(3)C_(1)` ways.
Hence total number of selections `= .^(3)C_(1)xx .^(3)C_(2)xx .^(3)C_(1)xx .^(3)C_(1)=81`
In each selection of six places, letters A, A, A, B,B, B can be arranged in `(6!)/(3! 3!)=20`
So, total number of ways `=81xx20=1620`
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