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02 May 2018, 08:33
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C
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Difficulty:
95%
(hard)
Question Stats:
26% (01:56) correct
74%
(01:56)
wrong
based on 160
sessions
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Given that the length of each side of a quadrilateral is a distinct integer and that the longest side is not greater than 7,How many different possible combinations of side lengths are there?
[A] 21
[B] 24
[C] 32
[D] 34
[E] 35
[A] 21
[B] 24
[C] 32
[D] 34
[E] 35
02 May 2018, 09:02
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That's really a nice question:
The concept used :
Since the length of each side of a quadrilateral is a distinct integer and that the longest side is not greater than 7, all four sides must be taken from the set {1, 2, 3, 4, 5, 6, 7}. How many different sets of four could be selected? \(7C4\)= 7*6*5/(3!) = 7*5 = 35 Of those 35, the only ones that don’t work for sides of a quadrilateral are the ones in which the sum of the three smallest sides are equal to or less than the longest side. These would be
1 + 2 + 3 < 7
1 + 2 + 3 = 6
1 + 2 + 4 = 7
Those are the only invalid combinations. The other 32 work.
Answer = (C)
The concept used :
like the triangle inequality theorem, the sum of the lengths of the shortest three sides of a quadrilateral must be longer than the longest side.
Since the length of each side of a quadrilateral is a distinct integer and that the longest side is not greater than 7, all four sides must be taken from the set {1, 2, 3, 4, 5, 6, 7}. How many different sets of four could be selected? \(7C4\)= 7*6*5/(3!) = 7*5 = 35 Of those 35, the only ones that don’t work for sides of a quadrilateral are the ones in which the sum of the three smallest sides are equal to or less than the longest side. These would be
1 + 2 + 3 < 7
1 + 2 + 3 = 6
1 + 2 + 4 = 7
Those are the only invalid combinations. The other 32 work.
Answer = (C)
General Discussion
02 May 2018, 09:13
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The simple proof of the concept used in this question :
"The sum of the lengths of the shortest three sides of a quadrilateral must be longer than the longest side."
Construct a random quadrilateral ABCD. 
gmatbusters2.jpg [ 12.4 KiB | Viewed 20321 times ]
Join any two vertices(Say AC)
Now from triangle inequality j+k>n
Also n+m>l
So, from the above two equations we have j+k+m > l
Hence,
It is analogous to triangle inequality theorem, which states that sum of two sides of a triangle is greater than third side.
"The sum of the lengths of the shortest three sides of a quadrilateral must be longer than the longest side."
Construct a random quadrilateral ABCD.
Attachment:
gmatbusters2.jpg [ 12.4 KiB | Viewed 20321 times ]
Now from triangle inequality j+k>n
Also n+m>l
So, from the above two equations we have j+k+m > l
Hence,
The sum of the lengths of the shortest three sides of a quadrilateral must be longer than the longest side.
It is analogous to triangle inequality theorem, which states that sum of two sides of a triangle is greater than third side.
