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Given that the length of each side of a quadrilateral is a distinct
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02 May 2018, 08:33
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20% (01:48) correct 80% (01:51) wrong based on 68 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
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Re: Given that the length of each side of a quadrilateral is a distinct
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02 May 2018, 09:02
That's really a nice question: 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)
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Given that the length of each side of a quadrilateral is a distinct
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02 May 2018, 09:13
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. Attachment:
gmatbusters2.jpg [ 12.4 KiB  Viewed 2781 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, 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.
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Re: Given that the length of each side of a quadrilateral is a distinct
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11 Jun 2018, 23:03
Hi, Sorry but why have you factored 1 + 2 + 3 two times? IMO the only combinations that dont satisfy the rule are 1,2,3 and 1,2,4. Thus 35  2 = 33 possible combinations. gmatbusters wrote: That's really a nice question: 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)



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Re: Given that the length of each side of a quadrilateral is a distinct
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12 Jun 2018, 00:29
The three sets of side lengths that do not work are: {1, 2, 3, 6} {1, 2, 3, 7} {1, 2, 4, 7} For each of these sets, the sum of the 3 smallest sides is NOT greater than the length of the longest side. Therefore, these 3 side lengths don’t work. ColonelRed wrote: Hi, Sorry but why have you factored 1 + 2 + 3 two times? IMO the only combinations that dont satisfy the rule are 1,2,3 and 1,2,4. Thus 35  2 = 33 possible combinations. gmatbusters wrote: That's really a nice question: 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) Posted from my mobile device
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Re: Given that the length of each side of a quadrilateral is a distinct
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