[email protected] wrote:133. Are all of the numbers in a certain list of 15 numbers
equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.

Target question: Are all 15 numbers equal?

Statement 1: The sum of all the numbers in the list is 60.
There are several possible scenarios that satisfy this statement. Here are two.
Case a: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, in which case all of the numbers are equal
Case b: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 7}, in which case all of the numbers are not equal
Statement 1 is NOT SUFFICIENT

Statement 2: The sum of any 3 numbers in the list is 12.
This is a very powerful statement.
One approach is to ask, "Can I construct a list of 15 numbers, in which the numbers are not all equal to 4 AND the sum of ANY 3 numbers is 12?"
If you try to create such a list, you'll find that it's impossible.

This, however, isn't a good proof. It's like "proving" there are no sasquatches by requiring others to find a sasquatch to prove you wrong.

Here's a more mathematical proof. It uses a technique called Reductio ad absurdum (not required for the GMAT!)

First let's assume that there is, indeed, a set in which the numbers are not all equal to 4 and the sum of ANY 3 numbers is 12.
Let's go on to say that a, b, c, . .. n and o are in this magical set.
This means that a+b+c must equal 12 (since they are 3 numbers in the set)
Also a+b+d must equal 12.
And a+b+e=12.
And a+b+f=12.
.
.
.

And a+b+o=12.

What does this tell us? It tells us that c = d = e = f = . . . = n = 0
We know this because when we add each of these numbers to a and b, we keep getting the sum of 12.

So, far we've proven that c, d, e . . . n and o are all equal.

Using similar logic, we can show that the same applies to a and b. In other words, we can show that a=d=e=f-. . . and b=d=e=f=. . .

So, even though we began by assuming that the numbers in the list are not all equal, we were forced to conclude that the numbers are all equal.

As such, our original assumption must be incorrect. In other words, there is no such set in which the numbers are not all equal to 4 AND the sum of ANY 3 numbers is 12.

Since such a list cannot exist, it must be the case that all 15 numbers are equal

So, statement 2 is SUFFICIENT

Answer = B

Phew . . . I'm calling it a day!

Cheers,
Brent

[email protected] wrote:133. Are all of the numbers in a certain list of 15 numbers
equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.

Please explain the authenticity of statement 2. not able to understand OG explanation.

Statement 1: The sum of all the numbers in the list is 60.
We can have different set of 15 numbers such that their sum is 60; NOT sufficient.

Statement 2: The sum of any 3 numbers in the list is 12.
This is only possible when all the numbers in the set are equal. Consider the case when we have a single different number, say a and other are equal, say b. Now the selection of any three numbers may result (a, b, b) and (b, b, b). Now according to the question, (a + b + b) and (b + b + b) both must be equal to 12. Which is only possible when a = b = 4.

This can shown for other combinations too. (Like 2 different, others equal etc); SUFFICIENT.

The correct answer is B.