Thread: Problem Solving 2 - eyebrow

Well i think ill just go ahead and make a separate thread for each question. Please could you post your worked out solutions and anything else which would help me understand how to solve the following problem. Thanks

Q2. An eyebrow 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 eyebrows is
a) 16 b)12 c)15 d) 24 e)18
Answer: A

Q2. An eyebrow 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 5 can be in the 2nd and 4th position since it's the largest number. when this occur.

when 5 is in the 2nd position, 4 must be in 4th position.

_ 5 _ 4 _

that's 3! = 6 ways
or vice versa = 6 x 2 =12 ways

or the 4th position could be occupied by 3, providing that the number 4 is not next to 3.

_ 5 _ 3 _

4 has two places to go. and the remaining has 1. hence 2 x 1 = 2 ways
and you can also have the alternate, where 3 sits in the 2nd position. hence 2 x 2 = 4 ways

Total = 4 +12 = 16 ways

Originally Posted by differentiate

Q2. An eyebrow 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 5 can be in the 2nd and 4th position since it's the largest number. when this occur.

when 5 is in the 2nd position, 4 must be in 4th position.

_ 5 _ 4 _

that's 3! = 6 ways
or vice versa = 6 x 2 =12 ways

or the 4th position could be occupied by 3, providing that the number 4 is not next to 3.

_ 5 _ 3 _

4 has two places to go. and the remaining has 1. hence 2 x 1 = 2 ways
and you can also have the alternate, where 3 sits in the 2nd position. hence 2 x 2 = 4 ways

Total = 4 +12 = 16 ways

Understood,
thanks