A 1×6 square unit rectangular grid has to be covered with six 1×1 square unit tiles. Three of the tiles are yellow, two are white, and one is blue. In how many ways can this task be done if
a
there are no restrictions?
6!/3!2!=60
Well, let's see. We have 6 tiles to cover a 1x6 grid. There are no restrictions, so we can place the tiles in any way we want.
Now, imagine for a moment that each tile has a unique personality. The yellow tiles are best friends and always want to be together. The white tiles, on the other hand, are a bit more independent and can be placed anywhere. And the blue tile? Well, he's a lone wolf and prefers to be on his own.
So, we need to find all the different combinations of placing the 6 tiles in the 1x6 grid. Let's break it down step by step:
1. Place the yellow tiles first. Since they want to be together, there are only 2 spots in the grid where they can go: either on the left or on the right. Once you decide where to place them, you're left with 4 empty spots.
2. Now it's time to place the white tiles. Since there are 2 white tiles, you have 4 options for the first white tile (the remaining empty spots), and 3 options for the second white tile (since one spot will already be taken by the first white tile).
3. Finally, you're left with the blue tile. Since he's a lone wolf, he can go in any of the remaining 2 spots.
So, by multiplying the number of options at each step together, we get: 2 (yellow) x 4 (first white tile) x 3 (second white tile) x 2 (blue) = 48.
Therefore, there are 48 different ways to cover the 1x6 grid with the 6 tiles, without any restrictions.
If there are no restrictions, we can calculate the number of ways to arrange the tiles by using the concept of permutations.
Since there are 6 tiles to be arranged, we have 6 options for the first tile, 5 options for the second tile, 4 options for the third tile, and so on. Hence, the total number of ways to arrange the tiles is given by:
6 × 5 × 4 × 3 × 2 × 1 = 720
Therefore, there are 720 ways to cover the 1×6 rectangular grid with the given tiles if there are no restrictions.
To solve this problem, we need to find out the total number of ways to arrange the six tiles on the 1×6 rectangular grid without any restrictions.
Since there are no restrictions, we can consider all the tiles as distinct. Let's label the yellow tiles as Y1, Y2, and Y3, the white tiles as W1 and W2, and the blue tile as B.
Now, we need to find the number of ways to arrange these six distinct tiles on the 1×6 rectangular grid. We can solve this using basic combinatorics.
To arrange the first tile, we have six options to choose from (Y1, Y2, Y3, W1, W2, or B). Once we place the first tile on the grid, we have five remaining options to choose from for the second tile. Similarly, after placing the second tile, we have four options for the third tile, then three options for the fourth, two options for the fifth, and only one option for the sixth tile.
To determine the total number of ways, we multiply all these options together:
6 × 5 × 4 × 3 × 2 × 1 = 720
Therefore, there are 720 different ways to cover the 1×6 rectangular grid with the six tiles when there are no restrictions.