2010 UNCO Math Contest II Problems/Problem 5

Problem

(a) In the $4 \times 4$ grid shown, four coins are randomly placed in different squares. What is the probability that no two coins lie in the same row or column?

$\begin{tabular}{|c|c|c|c|} \hline &&& \\ \hline &&& \\ \hline &&& \\ \hline &&& \\ \hline \end{tabular}$

(b) Generalize this to an $N \times N$ grid.

Solution

(a) Let us write the position of any coin as $(r,c)$ . Then, $0 \le r \le 3, 0 \le c \le 3$ . Therefore, there are $4 \times 4 = 16$ different positions for the first coin, $3 \times 3 = 9$ possibilities for the second coin, $2 \times 2 = 4$ possibilities for the third coin, and $1 \times 1 = 1$ possibility for the fourth coin. Together, there are $16 \times 9 \times 4 \times 1 = 576$ possible placements of the two coins.

(b)Using the same process above, the possible positions of a coin at $(r,c)$ , where $0 \le r \le N-1, 0 \le c \le N-1$ , can be counted as $(N-1)^2$ possibilities for the first coin, $(N-2)^2$ possibilities for the second coin, $(N-3)^2$ possibilities for the third coin, and $(N-4)^2$ possibilities for the fourth coin. Finally, there are $(N-1)^2 + (N-2)^2 + (N-3)^2 + (N-4)^2 = 4N^2 -20N + 30$ possibilities.

2010 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2010 UNCO Math Contest II Problems/Problem 5

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