Question 106
a) Consider the $4 \times 4$ board in the following figure. The letters and numbers were placed to help locate the small squares. For example, square $A 1$ is the one in the upper left corner and $D 4$ is the one in the lower right corner.
Show a way of choosing 12 small squares so that exactly 3 squares are chosen in each row and in each column.
b) Determine the number of different ways of choosing 12 small squares on the $4 \times 4$ board so that exactly 3 squares are chosen in each row and in each column.
c) Consider an $8 \times 8$ board with rows and columns labeled similarly to the previous board, but colored alternately black and white.
Determine the number of ways of choosing 56 small squares of the board so that all black squares are chosen and exactly 7 squares are chosen in each row and in each column.
Show a way of choosing 12 small squares so that exactly 3 squares are chosen in each row and in each column.
b) Determine the number of different ways of choosing 12 small squares on the $4 \times 4$ board so that exactly 3 squares are chosen in each row and in each column.
c) Consider an $8 \times 8$ board with rows and columns labeled similarly to the previous board, but colored alternately black and white.
Determine the number of ways of choosing 56 small squares of the board so that all black squares are chosen and exactly 7 squares are chosen in each row and in each column.