M23-22

Question

If two different numbers are randomly selected from set  \{1, 2, 3, 4, 5\} , what is the probability that the sum of the two numbers is greater than 4?

A.  \frac{3}{5}  
B.  \frac{7}{10}  
C.  \frac{4}{5}  
D.  \frac{9}{10}  
E.  \frac{19}{20}

Bunuel · Answer

Official Solution:

If two different numbers are randomly selected from set  \{1, 2, 3, 4, 5\} , what is the probability that the sum of the two numbers is greater than 4?

A.  \frac{3}{5}  
B.  \frac{7}{10}  
C.  \frac{4}{5}  
D.  \frac{9}{10}  
E.  \frac{19}{20}

Let's determine the probability of the opposite event first and then subtract that value from 1. The opposite event occurs when we select two different numbers such that their sum is  less than or equal to 4 .
 
The total number of outcomes is  C^2_5=10 , which represents choosing 2 different numbers from a set of 5.
 
The outcomes where the  sum \leq 4  are only (1,2) and (1,3), totaling 2 outcomes.
 
Thus,  P = 1 - \frac{2}{10} = \frac{4}{5} .

Answer: C

ajanpaul · Answer

How is it not B =7/10?

When you consider the case , how many pairs of numbers whose sum is greater than 4, you get

(1,4),(1,5).(2,4),(2,5),(2,3),(3,4),(3,5),(4,5) which is 7 outcomes out of total possible 10 outcomes.

Please explain.

Bunuel · Answer

It's not that hard to do the following:
1 - (1,4)
2 - (1,5)
3 - (2,4) 
4 - (2,5) 
5 - (2,3) 
6 - (3,4) 
7 - (3,5) 
8 - (4,5)

VeritasPrepErika · Answer

A quick way to test whether or not order matters is to try making the same selection in two different orders to see if you get a new result.

For example, if I am making a 3-person group from A, B, C, D, and E and I select A, then B, then C, I get the same group as when I select C, then B, then A. Same result = order doesn't matter = combination.

On the other hand, if I am selecting 1st, 2nd, and 3rd place from A, B, C, D, and E, and I select A, then B, then C, I get a different order than when I select C, then B, then A. Different result = order does matter = permutation.

Trying that here, if I am picking 2 numbers from {1,2,3,4,5}, and I pick 1, then 2, I get the same sum (3) as when I pick 2, then 1. Same result = order doesn't matter = combination.

While there are a few key words that indicate whether or not order matters, we don't always get them, so the best way to figure it out is to test it yourself.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

rmahe11 · Answer

I believe 3,2 is different than 2,3 although sum is greater than 4 but they are two technically different cases hence it has to be 14/20

Bunuel · Answer

The correct answer should be and indeed is 4/5. Eight pairs make the sum more than 4:

1 - (1,4)
2 - (1,5)
3 - (2,4)
4 - (2,5)
5 - (2,3)
6 - (3,4)
7 - (3,5)
8 - (4,5)

Then, there are 10 pairs possible. Hence, the probability is 8/10 = 4/5.

If you distinguish between (2, 3) and (3,2), though there is no reason to do that, then we have 16 pairs and 20 possible pairs, making the probability again equal to 16/20 = 4/5.

sb995 · Answer

I did not quite understand the solution. To clarify my understanding could you please explain why we are not taking 4 outcomes into consideration to take into account choosing (1,2), (1,3), (2,1) and (3,1)? Thank you

Bunuel · Answer

In the solution, we use the combination approach, which counts unordered pairs in both the numerator and denominator. So C(5,2) = 10 gives the total number of unordered pairs of two numbers from the set. In the numerator, we also have two unordered pairs, {1,2} and {1,3}. Because both numerator and denominator are expressed in terms of unordered pairs, the answer is consistent.

If instead we used ordered pairs, then both the numerator and denominator would be multiplied by 2, and those factors would cancel out, giving the same final result.

Helpmecat · Answer

I did not quite understand the question. the questions say the sum of 2 numbers is greater than 4 why the sum≤4 in the explanation?

Bunuel · Answer

Because the solution uses the opposite event. Instead of counting cases where the sum > 4 directly, the solution counts cases where the sum is ≤ 4, and then subtracts from 1.

M23-22

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards