How many different necklaces can we make from n beads of different colors? Consider two necklaces the same if (like in a circular arrangement) one can be obtained from the other via rotation or if (unlike in a circular arrangement) one can be obtained from the other via flipping the necklace over.

Step-by-step explanation:

Let us consider n beads of different colors, arranged in a line. If the beads were to be arranged in a straight line, there are n! ways to do this.

Now, if the beads were to be in a circular arrangement, a pattern will repeat n times.

So the number of different circular arrangements can be obtained by dividing by n, such that we get =

The patterns can be obtained via rotating either clockwise or anticlockwise, therefore 2 ways.  So we can divide the total by 2.

Hence, the different number of necklaces which we can make from n beads of different colors is

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step-by-step explanation:

answer

the correct answer is

explanation

the given function is

the excluded values are obtained by solving:

we factor to obtain:

the excluded values are