Juana mailed at least one letter on each day last week. Was the total

The number of letters was positive on each day.

We need to answer the question:

Is ℓ₁ + ℓ₂ + ℓ₃ + ℓ₄ + ℓ₅ + ℓ₆ + ℓ₇ > 27 ?

Statement One Alone:

=> Juana mailed fewer than 8 letters on each of the 7 days.

Let’s imagine the daily numbers of letters in chronological order.

If {7, 7, 7, 7, 7, 7, 7}, then the answer is Yes.

Whereas, if {1, 1, 1, 1, 1, 1, 1}, then the answer is No.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> Juana mailed a different number of letters on any two of the 7 days.

If {1, 2, 3, 4, 5, 6, 7}, then the sum is 28, and the answer is Yes. Since 28 is the least possible sum for different numbers of letters, we have a definite Yes answer.

Statement two is sufficient.

Answer: B

Could you explain more regarding the meaning of "on any two of the 7 days" in Statement (2)

What if I say
"on any three of the 7 days"
Does this has any meaning?
or just the same as
"on any two of the 7 days"?

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sanebeyondone ­No, "a different number on any three of the 7 days" has no meaning, and wouldn't be used on the GMAT or normal English.

(2) Juana mailed a different number of letters on any two of the 7 days.

@Matsu12  "a different number on ANY two ____" means that none of the numbers are the same. This has the same meaning as the wording we may see in other problems, "7 different integers are selected".

Therefore, the minimum possible sum is the evenly spaced set, 1+2+3+4+5+6+7, since the stem says she mailed "at least one letter on each day".
The fastest way to calculate a sum of consecutive integers is S = AN. 
For an evenly spaced set, A = Average = Median = ((First + Last) / 2)
N = the number of numbers
S = AN = 4*7 = 28, so statement (2) is sufficient.

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Total letters mailed is more than 7 (at least 1 letter each day)
Question: Is Total > 27 ?

(1) Juana mailed fewer than 8 letters on each of the 7 days.

This means she mailed at most 7 letters each day. So max she mailed 49 letters total. We don't know whether total was less or more than 27.
Not Sufficient

(2) Juana mailed a different number of letters on any two of the 7 days.

So minimum number of letters she mailed is 1 + 2 + 3 +... + 7
Sum of n consecutive integers starting from 1 is given by \(\frac{n(n+1)}{2} = \frac{7*8}{2} = 28\)
So she mailed at least 28 letters. Sufficient to say that Yes, total is more than 27. 

Answer (B)
 

­Statement (2) - Any two of the 7 days could mean same number of letters for 5 days and different numbers for the rest of the days. Why is it different for all the days?

"­mailed a diferent number of letters on any two of the 7 days" means if you pick ANY 2 days of the 7, the number of letters will be different. Hence number of letters has to be different on all days. 

"mailed a diferent number of letters on two of the days" means that on some specific 2 days the number of letters was different. ­We don't know about other days.

any two of seven days means ?

­The phrase "any two of the seven days" means that Juana mailed a different number of letters on each day, so no two days had the same count of mailed letters.

(2) Juana mailed "a different number of letters on any two of the 7 days."

Who speaks like this.