Solution:

The password that opens the chamber is 65292.

Solution:

There can be various solutions. Here are a few that we came up with: 1 + 2 + 4/6 + 5 + 7/3 + 89 = 100
12 + 6/3 + 75 + 8/4 + 9 = 100
1/6 + 2 + 4/8 + 7/3 + 95 = 100
24 + 57 + 6/3 + 8 + 9/1 = 100
36 + 4/8 + 51 + 7/2 + 9 = 100
3/2 + 5 + 6/8 + 7/4 + 91 = 100

Solution:

India has the highest chances of winning the war (47/90)
Missile Firing order:
Chances of Remaining Unhit - Pakistan:
Chances of Remaining Unhit - Srilanka:
Chances of Remaining Unhit - India

Pakistan - Srilanka - India
1/2 0 1/2

Pakistan - India - Srilanka
1/2 0 1/2

India - Pakistan - Srilanka
1/10 16/45 49/90

Srilanka- Pakistan - India
1/2 0 1/2

Srilanka - India - Pakistan
1/10 16/45 49/90

India - Srilanka - Pakistan
1/10 16/45 49/90

Total chances of being unhit (sum of the probabilities divided by 6):
27/90 16/90 47/90

Solution:

45

1/5 of 45 = 9
9 + 45 = 54

Solution:

There is a chance of 1/3 to be its best in all the shots. The reason for this is that the shots do not depend on the shot that is made earlier. The chance for the third shot to be its best is 1 minus the chance that the last shot is the best shot. Therefore the answer for this question is 2/3.
There are also other possibilities. There are three arrows A, B, and C. the B arrow is away from the arrow A. so after the shot A the following will be the outcome or the possibility with the shots.
Best shots: A, A, C
Worst shots: C, B, B
From all the three outcomes, the arrow C is worse than the arrow A.

Solution:

7 Races.

Solution:

{a +b +abs (a-b)}/2
It is shown that {a +b-abs (a-b)}/2 is < {a, b}

Solution:

Charlie will be able to buy 22 chocolates.

Explanation:
The cost of each chocolate is rs.3 and Charlie has rs.45. He buys 15 chocolates with them. Since there is an offer and the scheme, he returns 15 wrappers and he gets 5 chocolates in return for free. Again he returns 3 wrappers and gets another chocolate free. Again he uses that one wrapper with the other two and gets another chocolate. So all together it is 22 chocolates.

Solution:

The best way to increase the probability of winning is to place one yellow ball in the first bowl and all the other balls in the second bowl.

Now, you have a 1/2 probability of choosing the bowl with just one yellow ball. If you choose that, you win.

If you choose the other bowl, it has 49 yellow balls and 50 white balls. You still have almost 1/2 probability of picking a yellow ball out of it and winning.

Solution:

Both S and T are of same height because they are the same people.

Suppose that each person is marked with their respective number which also denotes their height.
Now when the longest from each row is called out, they will be 50, 100, 150 ... 500. Shortest among them will be 50 aka S.
When the shortest from each colum is called out, they will be 1, 2, 3, ....50. Longest among them will be 50 aka T.

Thus, they both are same person.