If P is the product of n distinct integers, is P<0?

_____________________
Edited the OA. Thank you.

We know that numbers are consecutive integers.

For statement 2,

If 5 values out of n are negative than, the remaining values will be equal to or more than 0. Therefore, if the value of n is 5 then P will be negative; otherwise, P will be 0 in all other cases.
For ex. -5, -4, -3, -2, -1 then P IS NEGATIVE.
-5, -4, -3, -2, -1, 0 then P is 0.
-5. -4, -3, -2, -1, 0, 1 Then P is O.

The main problem is that we are not told whether n is 5.

quote="IanStewart"]
If the OA is B, there's something wrong with the question. If Statement 1 is true, then either the set contains a small number of consecutive integers, and every value is positive, so their product is positive, or the set is larger and the lower values in the set extend to zero or past zero into negative values. But if it's a set of consecutive integers, then the set would then contain zero, so the product of the values in the set would be zero. The product could never be negative, so Statement 1 is sufficient to give a "no" answer to the question.

Statement 2 is not sufficient, because if we have five negative values, and the rest of the values are positive, the product is negative, but if we have five negative values and we have zero in the set, the product is zero. So we can't be sure if the product is negative.[/quote]

According to your explanation,
terms can be -5,-4,-3,-2,-1,0,1,2,3,4,5- product=0

But in your sequence, the median is not 4.5.
We can understand this as below.
Seq.1= 2, 3, 4, 5, 6 7 >>>>P= POSITIVE
Seq.2= 1, 2, 3, 4, 5, 6, 7, 8>>>>>p= zero
Seq.3 = -1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 =>>>>P=ZER0.

so P will never be negative.

Considering Option 1, Given

n distinct integers are consecutive numbers and their median is 4.5

considering x(as one integer) & y(as another integer)

(x+y)/2 =4.5 ==> (x+y) = 9 ==> (x+x+1) = 9 ==> x = 4

Therefore one integer is 4 & one integer is 5

the sequences are beyond 0 i.e. -4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13 - Product is 0

the sequences are after 0 i.e. 1,2,3,4,5,6,7,8- Product > 0

No chance for P to be less than 0

From Option 2

if 5 integers are negative, as per the example above P can be -ve or 0

So we can opt option A

I have used a strategy to solve the question. The detailed solution is provided in the video below. I hope this helps.