Solution:

0.1667 or 16.67%

When rolling two fair six-sided dice, there are 36 possible outcomes (6 outcomes for the first die multiplied by 6 outcomes for the second die).

To calculate the probability of rolling a sum of 7, we need to determine the number of combinations that result in a sum of 7. These combinations are:

1 + 6
2 + 5
3 + 4
4 + 3
5 + 2
6 + 1

So, there are 6 favorable outcomes that result in a sum of 7.

The probability of rolling a sum of 7 is calculated by dividing the number of favorable outcomes by the total possible outcomes:

Probability = (Number of Favorable Outcomes) / (Total Possible Outcomes) = 6 / 36 = 1/6 ≈ 0.1667

Therefore, the probability of rolling a sum of 7 with two fair six-sided dice is approximately 0.1667 or 16.67%.

Solution:

Think deeply about the door number 56, people will visit it for every divisor it has. So 56 has 1 & 56, 2 & 28, 4 & 14, 7 & 8. So on pass 1, the 1st person will open the door; pass 2, 2nd one will close it; pass 4, open; pass 7, close; pass 8, open; pass 14, close; pass 28, open; pass 56, close.
Thus we can say that the door will just end up back in its original state for each pair of divisor. But what about the cases in which the pair of divisor has analogous number for example door number 16? 16 has the divisors 1 & 16, 2 & 8, 4&4. But 4 is recurrent because 16 is a perfect square, so you will only visit door number 16, on pass 1, 2, 4, 8 and 16… leaving it open at the end. So only perfect square doors will remain open at the end.

Solution:

step 3 is wrong

Solution:

192 021

If you look at it as two-digit numbers you'll see:10 11 12 13 14 15 16 17 18 19 20 21

Solution:

It has been clearly mentioned that all the boxes are labeled incorrectly. If he opens the Box3, then he will get either Dark Chocolates or Milk Chocolates as it is labeled incorrectly. Let us suppose he finds Dark Chocolates in there. Now since all are labeled incorrectly, Box B A must contain Milk Chocolates and Box B must contain Milk Chocolates and Dark Chocolates.

Solution:

It can be done as shown in the picture below.

Solution:

They used the planks in the manner that is shown in the figure.

Solution:

Upon reaching each room, he found out that the window facing the dead body was closed. He had to open the window himself. This tells that there was somebody else with him who closed the window. If he would have committed suicide, at least one window would have been open.

Solution:

For this answer is 3^0, 3^1, 3^2... That is 1,3,9,27,81,243 and 729.

Solution:

It can be solve as shown below.

Solution:

Here is a way to do it. We dont know if it can be done using some other arrangement or not. If you know, do send us.

4 | 3 | 8
9 | 5 | 1
2 | 7 | 6