probability & geometry

Question

Can you please help me with this?
3. If three different integers are selected at random from the integers 1 through 8, what is the probability that the three selected integers can be the side lengths of a triangle?

A. 11/28

B. 27/56

C. 1/2

D. 4/7

E. 5/8

Rich.C@EMPOWERgmat.com · Answer

Hi amrabdelnaby,

I'm going to give you some 'hints' so that you can try this question again on your own:

1) This question is based on the Triangle Inequality Theorem - the idea that if you add up any 2 sides of a triangle, THAT sum MUST be greater than the length of the third side (for example, a triangle can have sides of 3, 4 and 5 since 3+4 > 5, 3+5 > 4 and 4+5 > 3....but a triangle CANNOT have sides of 1, 2 and 4 since 1+2 is NOT > 4).

2) Since we're choosing 3 DIFFERENT integers, and the 'order' of the 3 integers does not matter, we can use the Combination formula to figure out the TOTAL number of possible groups of 3 integers.

3) You will probably find it easier to find the number of groups that do NOT create an actual triangle than the number that DO.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

Amrabdelnaby · Answer

Hi Rich,

So here is how i solved, yet it took from me some time. I hope you can give me an easier way/short cut to solve it.

Step 1: all possible ways to choose 3 from 8 is 8C3 = 56
Step 2: break rule: I listed the 8 numbers as shown: 1 2 3 4 5 6 7 and determined that the first number has 7 possibilities of not making a triangle, the second has 6, the third has 5 and so on until reaching the seventh which has one possibility.

Step 3: add all possibilities: 7+6+5+4+3+2+1= 3.5x8 = 28

Step 4= 52/52 - 28/52 = 22/52 = 11/26.

Is there another way to solve it faster or is this the only way?

Please advise.. thanks

[email protected] wrote:Hi amrabdelnaby,

I'm going to give you some 'hints' so that you can try this question again on your own:

1) This question is based on the Triangle Inequality Theorem - the idea that if you add up any 2 sides of a triangle, THAT sum MUST be greater than the length of the third side (for example, a triangle can have sides of 3, 4 and 5 since 3+4 > 5, 3+5 > 4 and 4+5 > 3....but a triangle CANNOT have sides of 1, 2 and 4 since 1+2 is NOT > 4).

2) Since we're choosing 3 DIFFERENT integers, and the 'order' of the 3 integers does not matter, we can use the Combination formula to figure out the TOTAL number of possible groups of 3 integers.

3) You will probably find it easier to find the number of groups that do NOT create an actual triangle than the number that DO.

GMAT assassins aren't born, they're made,
Rich

Rich.C@EMPOWERgmat.com · Answer

Hi amrabdelnaby,

Certain questions on Test Day are designed to take longer than others to solve (some of them could take you 3 minutes to solve - and that's if you KNOW what you're doing), so if this question took you 2-3 minutes to solve, then that's fine. If it took you 4-5+ minutes to solve, then you probably weren't doing any work for most of that time. You likely won't find an 'easier' solution than the one that you used though, so you shouldn't overthink this one question.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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