Problem Understanding: Placing Coins on N x N Board

The question asks for the number of distinct ways to arrange N coins on an NxN grid (board) such that there is precisely one coin in each row and exactly one coin in each column. This is equivalent to finding the number of permutations of N items.

Step-by-Step Coin Placement Strategy

We can determine the number of ways by considering the placement choices for each row sequentially:

• Row 1: There are N columns available to place the first coin. So, there are N choices.
• Row 2: One column is already occupied. Thus, there are N-1 remaining column choices for the second coin.
• Row 3: Two columns are occupied. There are N-2 column choices left for the third coin.
• Continuing this pattern: For each subsequent row, the number of available column choices decreases by one.
• Row N: For the last row, there is only 1 column choice left.

Calculating Total Placement Ways

To find the total number of ways, we multiply the number of choices for each row, according to the multiplication principle of counting:

Total Ways = (Choices for Row 1) $\times$ (Choices for Row 2) $\times$ ... $\times$ (Choices for Row N)

Total Ways = $N \times (N-1) \times (N-2) \times \dots \times 2 \times 1$

Result: Factorial Representation

The product $N \times (N-1) \times (N-2) \times \dots \times 2 \times 1$ is mathematically defined as the factorial of N, denoted by $N!$.

The notation $N(N-1)/(N-2)\dots 2 \times 1$ in Option B represents this factorial calculation, $N!$.

Therefore, there are exactly $N!$ ways to place the N coins on the N x N board satisfying the given conditions.