Solution:

The probability is 5/36.

Event = {(1,5) , (2,4), (3,3), (4,2), (5,1)}
Therefore, P (getting sum = 6) = 5/36

Solution:

Never

Explanation:
In the question, the distance of x miles is given at a particular moment. The middle car is running at that time at 50 MPH. The distance will keep increasing with time as three are running at different speeds. It will increase by x miles every hour till a certain point of time and then it will start decreasing till the fastest car meets the slowest again.

As per question, all the cars stop then. Thus, the distance will never be x miles again.

Solution:

So that the following plan can be followed, let us number the coins from 1 to 12. For the first weighing let us put on the left pan coins 1,2,3,4 and on the right pan coins 5,6,7,8.

There are two possibilities. Either they balance, or they don't. If they balance, then the different coin is in the group 9,10,11,12. So for our second one possibility is to weigh 9,10,11 against 1,2,3

(1) They balance, in which case you know 12 is the different coin, and you just weigh it against any other to determine whether it is heavy or light.
(2) 9,10,11 is heavy. In this case, you know that the different coin is 9, 10, or 11, and that that coin is heavy. Simply weigh 9 against 10; if they balance, 11 is the heavy coin. If not, the heavier one is the heavy coin.
(3) 9,10,11 is light. Proceed as in the step above, but the coin you're looking for is the light one.

That was the easy part.

What if the first weighing 1,2,3,4 vs 5,6,7,8 does not balance? Then any one of these coins could be the different coin. Now, in order to proceed, we must keep track of which side is heavy for each of the following weighings.

Suppose that 5,6,7,8 is the heavy side. We now weigh 1,5,6 against 2,7,8. If they balance, then the different coin is either 3 or 4. Weigh 4 against 9, a known good coin. If they balance then the different coin is 3, otherwise it is 4. The direction of the tilts can tell us whwther the offending coin is heavier or lighter.

Now, if 1,5,6 vs 2,7,8 does not balance, and 2,7,8 is the heavy side, then either 7 or 8 is a different, heavy coin, or 1 is a different, light coin.

For the third weighing, weigh 7 against 8. Whichever side is heavy is the different coin. If they balance, then 1 is the different coin. Should the weighing of 1,5, 6 vs 2,7,8 show 1,5,6 to be the heavy side, then either 5 or 6 is a different heavy coin or 2 is a light different coin. Weigh 5 against 6. The heavier one is the different coin. If they balance, then 2 is a different light coin.

Solution:

Ms. Britney did it.

We will take up every person and analyze the statements over all of them one by one.

Case 1: Ms. Britney did it, then
Statement 1 => true
Statement 2 => false (as Rihanna did not do it)
Statement 3 => false (as Ms. Britney is lying)
Statement 4 => false (as Ms. Gwyneth is lying)
Our first guess itself was right but let us confirm our analysis by analyzing the statements over others.

Case 2: Ms. Rihanna did it, then
Statement 1 => false (as Ms. Britney did not do it)
Statement 2 => true
Statement 3 => true (as Ms. Britney is not lying)
As per the conditions, three statements had to be wrong but here, we have two statements right which clearly contradict the condition.

Case 3: Ms. Gwyneth did it, then
Statement 1 => false (as Ms. Britney did not do it)
Statement 2 => false (as Ms. Rihanna did not do it)
Statement 3 => false (as Ms. Britney is lying)
Statement 4 => false (as Ms. Gwyneth is lying)
She did not do it as all the statements are false.

Case 4: Ms. Yoon did it, then
Statement 1 => false (as Ms. Britney did not do it)
Statement 2 => false (as Ms. Rihanna did not do it)
Statement 3 => false (as Ms. Britney is lying)
Statement 4 => false (as Ms. Gwyneth is lying)
All four statements are false in her case as well thus she did not do it.

Therefore we have proved that Ms. Britney did it.

Solution:

A
D
C
A
B

Solution:

Solution is shown in image below

Solution:

It is not possible to guess only three fruits correctly: the fourth fruit is then correct too! So nobody has guessed three fruits correctly and 123-43-39-31 = 10 people have guessed four fruits correctly.

Solution:

I am Spice

Explanation:
Spice is hot if you remove first two alphabets then it will become "ice", which is really cold.

Solution:

Sam : 2
Dean : 5
Castiel : 7

Age diff
7 - 2 = '5' is prime
7 - 5 = '2' is prime
5 - 2 = '3' is prime

Solution:

Pages 19, 20, 23, 37, 38, & 42 will also be missing

Solution:

Step 5 is invalid, because we are dividing by (x-y), and since x=y, we are thus dividing by 0. This is an invalid mathematical operation (division by 0), and so by not following basic mathematical rules