SOLUTION: An eyebrow number is an arrangement of the numbers 1, 2, 3, 4 and 5 such that the second and fourth numbers are each bigger than both their immediate neighbours. For example, (1, 3

Question 982390: An eyebrow number is an arrangement of the numbers 1, 2, 3, 4 and 5 such that the second and fourth numbers are each bigger than both their immediate neighbours. For example, (1, 3, 2, 5, 4) is an eyebrow and (1, 3, 4, 5, 2) is not. The number of eyebrow is
A. 16
B. 12
C. 15
D. 24
E. 18
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
I would say there are eyebrow numbers.
Here is why:
The lesser of the second and fourth numbers must be 3 or greater,
because it must be greater than two immediate neighbors,
and those immediate neighbors could not be smaller than 1 and 2.
If 3 is in one of those two special places (the second or the fourth place),
numbers 1 and 2 must be flanking it,
leaving numbers 4 and 5 to fill the other two places,
and number 4 must be at one end, next to number 5,
which would be in the other of those two special places (the second or the fourth place).
So the eyebrow arrangement would look like one of these patterns:
_ 3 _ 5 4 or 4 5 _ 3 _ .
The numbers 1 and 2 go in the blank places, in that order, or in the reverse order.
With choices for the order of 1 and 2,
and patterns to fill,
we get such arrangements.
If the lesser of the second and fourth numbers is not 3, it must be 4.
In that case the possible patterns are
_ 4 _ 5 _ or _ 5 _ 4 _ ,
each with blank spaces to be filled by the numbers 1, 2, and 3 in any order.
There are possible orders for the numbers 1, 2, and 3,
which applied to each of those patterns,
gives us eyebrow arrangements.
The total number of eyebrow arrangements is then

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