Necklace (combinatorics)
A necklace is like a pretty circle made up of different colored beads. How many different necklaces can we make if we have a bunch of beads in different colors and we can slide the beads around on the string to try out different combinations?
Let's say we have beads in four different colors: red, blue, green, and yellow. We can arrange them in many different ways to make different necklaces. We could put all the beads in a row and thread them onto the string, but that would make a boring necklace! Instead, we want to make a circular necklace with the beads.
First, let's see how many different combinations of beads we can make using just one bead. We can choose from any of the four colors, so we have four options. Now, let's add a second bead to the necklace. We still have four options for the second bead, but we have to remember that when we put it on the string, it can be next to any of the four sides of the first bead. So, there are actually four different ways to put the second bead next to the first bead.
If we keep adding beads to the necklace, we have to take into account all the ways the previous beads could be arranged. For example, if we have three beads on the necklace, we have to remember that the third bead could be next to any of the four sides of the second bead. But, we also have to remember that the second bead could be next to any of the four sides of the first bead. So, there are actually 4 x 4 = 16 different ways to arrange the first three beads.
Now, let's say we want to make a necklace with four beads. Using the same logic as above, we can see that we have 4 x 4 x 4 = 64 different ways to arrange the four beads. But, wait! This answer is counting some necklaces more than once. For example, if we have a necklace with the bead sequence red-blue-green-yellow, we could also make the same necklace by sliding the beads around and starting with the yellow bead. So, we really only have 64/4 = 16 different necklaces.
So, in summary, if we have four different colored beads and we want to make a circular necklace with them, we have 16 different combinations we can make. We can use this same logic with different numbers of beads and colors to find out how many different necklaces we can make with different combinations.
Let's say we have beads in four different colors: red, blue, green, and yellow. We can arrange them in many different ways to make different necklaces. We could put all the beads in a row and thread them onto the string, but that would make a boring necklace! Instead, we want to make a circular necklace with the beads.
First, let's see how many different combinations of beads we can make using just one bead. We can choose from any of the four colors, so we have four options. Now, let's add a second bead to the necklace. We still have four options for the second bead, but we have to remember that when we put it on the string, it can be next to any of the four sides of the first bead. So, there are actually four different ways to put the second bead next to the first bead.
If we keep adding beads to the necklace, we have to take into account all the ways the previous beads could be arranged. For example, if we have three beads on the necklace, we have to remember that the third bead could be next to any of the four sides of the second bead. But, we also have to remember that the second bead could be next to any of the four sides of the first bead. So, there are actually 4 x 4 = 16 different ways to arrange the first three beads.
Now, let's say we want to make a necklace with four beads. Using the same logic as above, we can see that we have 4 x 4 x 4 = 64 different ways to arrange the four beads. But, wait! This answer is counting some necklaces more than once. For example, if we have a necklace with the bead sequence red-blue-green-yellow, we could also make the same necklace by sliding the beads around and starting with the yellow bead. So, we really only have 64/4 = 16 different necklaces.
So, in summary, if we have four different colored beads and we want to make a circular necklace with them, we have 16 different combinations we can make. We can use this same logic with different numbers of beads and colors to find out how many different necklaces we can make with different combinations.
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