Brent@GMATPrepNow wrote:If 3 different integers are randomly selected from the integers from 1 to 12 inclusive, what is the probability that a triangle can be constructed so that its 3 sides are the lengths of the 3 selected numbers?
A) 3/8
B) 7/18
C) 19/44
D) 39/88
E) 11/24
Answer:
C
Source:
www.gmatprepnow.com
Difficulty level: 700+
Cheers,
Brent
I specifically created this question to illustrate the importance of
calculating the denominator first (when using counting techniques to solve a probability question)
There are two reasons why you should calculate the denominator first:
1) The denominator is almost always easier to calculate than the numerator, and while calculating the denominator, you may gain some insight into how to calculate the numerator.
2) Once you know the denominator, you can use this to eliminate answer choices. So, even if you don't know the correct answer, you can still eliminate some answers and increase your chances of guessing correctly.
If you're lucky, you can eliminate 4 of the 5 answer choices, as you have done above!!
For this question, P(a triangle can be constructed with the 3 selected lengths) = (
number of triangles with 3 lengths from 1 to 12 inclusive)/(
TOTAL number of ways to select 3 numbers)
Let's calculate the denominator.
TOTAL number of ways to select 3 numbers
Since the order in which we select the 3 numbers doesn't matter, we can use COMBINATIONS.
We can select 3 numbers from 12 numbers in 12C3 ways.
12C3 = (12)(11)(10)/(3)(2)(1) =
220 ways
So, the correct answer will either be in the form k/220, OR some equivalent fraction in which the denominator is a FACTOR of 220.
For example,
IF we calculate the numerator and get 110, then the answer = 110/220 = 1/2 (notice that 2 is a FACTOR of 220)
IF we calculate the numerator and get 15, then the answer = 15/220 = 3/44 (notice that 44 is a FACTOR of 220) And so on.
When we check the answer choices, we see that only one answer choice (C) has a denominator that's a FACTOR of 440.
So, C must be the correct answer.
We can answer the question without having to calculate the numerator (which is a time-consuming task)
On test day, it's unlikely that this technique will allow you to eliminate 4 answer choices. HOWEVER, if you're pressed for time, or you can't calculate the numerator, this technique may allow you to eliminate some of the answer choices and increase your likelihood of a correct guess.
Cheers,
Brent
