In how many ways can cells in a $3 \times 3$ grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown.
To solve this problem, we need to determine in how many ways we can shade one cell in each row and each column of a \(3 \times 3\) grid. This is equivalent to finding the number of permutations of three distinct items, which corresponds to the number of ways to arrange the numbers 1, 2, and 3.
- Consider the first row: We can choose any of the 3 cells to shade. This gives us 3 options.
- For the second row: We need to choose a different cell from the one shaded in the first row, to ensure each column has exactly one shaded cell. This gives us 2 options.
- For the third row: We are left with only 1 choice, as the cells in the same column as the previously shaded cells cannot be chosen.
Therefore, the total number of ways to achieve the shading is the product of the number of choices available for each row:
\(3 \times 2 \times 1 = 6\)
Hence, there are 6 ways to shade the cells such that each row and each column has exactly one shaded cell.
The correct answer is therefore 6.
Important Questions from Permutations
How many words can be formed with the letters of the word 'POSTMAN', if every word begins with T and ends with M?
- Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?
- The number of 'three-digit numbers' that can be formed using the digits from 1 to 9 without the repetition of each digit is ________.
- The number of ways in which the letters in the word MINING can be arranged is
- A box containing 10 identical compartments has 6 red balls and 2 blue balls. If each compartment can hold only one ball, then the number of different possible arrangements are
