There are 4 kinds of colored beads( r, g, b, y) to seal onto the 6 vertices of a regular hexagon, how many solutions are there?
To find the number of solutions, we can approach the problem using a technique called counting.
Step 1: Selecting the color for each vertex.
For each vertex of the hexagon, we have 4 choices of colored beads: r, g, b, or y. Since there are 6 vertices, we can consider the choices for each vertex independently.
Step 2: Counting the possibilities.
For the first vertex, we have 4 choices of colored beads.
Similarly, for the second vertex, we have 4 choices as well.
Continuing this pattern, for the third, fourth, fifth, and sixth vertices, we also have 4 choices each.
To find the total number of solutions, we multiply the number of choices for each vertex together. In this case, since each vertex has the same number of choices, we can simply raise the number of choices to the power of the number of vertices.
Total number of solutions = (Number of choices for each vertex)^(Number of vertices)
In our case, there are 4 choices for each of the 6 vertices.
Total number of solutions = 4^6
= 4 * 4 * 4 * 4 * 4 * 4
= 4096
Therefore, there are 4096 different solutions to seal the colored beads onto the 6 vertices of the regular hexagon.