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In how many ways can 5 different keys be placed on a keychain? (Two arrangements are considered the same if one can be rotated or flipped to be identical to the other.)

Jun 25, 2021

#1
+151
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if we were to arrange the keys in a row it would be factorial 5, and on a circle, it would be factorial 4, but since it's on a keychain we have to divide by two

thus we get $\frac{4!}{2} \ \ \Rightarrow \ \ \ \frac{24}{2} = 12$ different ways.

Jun 25, 2021

#1
+151
+2

if we were to arrange the keys in a row it would be factorial 5, and on a circle, it would be factorial 4, but since it's on a keychain we have to divide by two

thus we get $\frac{4!}{2} \ \ \Rightarrow \ \ \ \frac{24}{2} = 12$ different ways.