Problem number 27 states, in the 4x4 table, some cells must be painted black. The numbers next to and below the table show how many cells in that row or column must be black. And how many ways can this table be painted? So let's start off here with our 4x4 table. We can immediately notice that there will be nothing in the second row or the second column because the numbers adjacent to those rows and columns are zero. In order to solve this problem, let's go column by column making assumptions until we've made all the assumptions we can and then count up all of the possibilities. So for example, for the first column, let's assume that it's going to be the top two squares that are going to be shaded. So now let's go on to the third column. There are three ways that we could shade this next column. This is how they would look. All of these would be valid. From here, there's only one way to finish off each of these scenarios, and that would be like this. So if we initially assume the top two available squares in the first column are shaded, then we have three ways to shade the table. Next let's assume the top most and bottom most squares are shaded in the first column. There's only one possible move for the next column since the bottom row can only have one shaded square. So let's shade the top two. There's only one move from here in the last column, and that is this one. So in total, there's only one way to shade the table when we assume the top and bottom square is shaded for the first column. Now lastly, let's assume the bottom two are shaded. There's only one possible move for the next column since the bottom row can only have one shaded square. And last, the top square needs to be shaded in the first column because that is the only row missing a number. So there is only one way to shade the table if the bottom two squares are shaded in the first column. So in total, there are five ways to shade this table. Three if the top two are shaded, one if the bottom two are shaded, and another one if the top and bottom is shaded in the first column. So in total, that's five different ways to shade the table. So the question asks us, in how many ways can this table be painted? The answer is five. Letter D.

4x4 table

cell shading

row and column numbers

distinct configurations

painting methods