Given that the length of each side of a quadrilateral is a distinct integer and that the longest side is not greater tha

Expert replies

Previous Topic Next Topic

VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Sat Jul 11, 2020 11:23 pm

Timer

00:00

Answers

Stats

Difficulty—

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
24
[C] 32
[D] 34
[E] 35

[spoiler]OA=C[/spoiler]

Source: Magoosh

Like this post

Source: Beat The GMAT — Problem Solving | Sat Jul 11, 2020 11:23 pm

GMAT/MBA Expert

Scott@TargetTestPrep: GMAT Instructor; Posts: 8152; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: Given that the length of each side of a quadrilateral is a distinct integer and that the longest side is not greater

by Scott@TargetTestPrep » Fri Jul 23, 2021 6:40 am

VJesus12 wrote: ↑
Sat Jul 11, 2020 11:23 pm
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
24
[C] 32
[D] 34
[E] 35

[spoiler]OA=C[/spoiler]

Solution:

Recall that in a quadrilateral, the sum of three sides must be greater than the remaining side.

Let’s first determine the number of ways to pick four distinct positive integers not greater than 7; i.e. from among 1, 2, 3, 4, 5, 6, and 7. Since there are 7 integers and we are choosing 4 of them, there are 7C4 = 7!/(4!*3!) = (7x6x5)/(3x2) = 35 ways to make this choice.

Among these 35 possibilities, some of them won’t satisfy the rule we mentioned above. To determine the invalid choices, let’s choose three sides to be as small as possible.

If the three smallest sides are 1, 2 and 3, then we see that the fourth side cannot equal 1 + 2 + 3 = 6 or any length greater than 6. We see that there are 2 invalid choices (which are 1-2-3-6 and 1-2-3-7).

If the three smallest sides are 1, 2 and 4, then we see that the fourth side cannot equal 1 + 2 + 4 = 7 or any length greater than 7. We see that there is 1 invalid choice (which is 1-2-4-7).

If the three smallest sides are not 1-2-3 or 1-2-4, then the sum of the three smallest sides is at least 8 and there are no invalid choices.

Since there are 35 possibilities in total and 3 of them are invalid, the number of possible combinations of side lengths is 35 - 3 = 32.

Answer: C

Scott Woodbury-Stewart
Founder and CEO
[email protected]

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews

Like this post

Post new topic Post Reply

• Page 1 of 1

GMAT Course Reviews

560 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (560 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review