Solution:

If you have come up with the fact that the bugs can avoid the collision only when they decide to move in the same direction be it clockwise or anti clockwise, you have already solved half of it. In any other way, the bugs will definitely run into each other.

Now the bugs can move either clockwise or anti-clockwise. The probability of choosing a specific direction is then 1/2.

We will utilize the simple equation of probability to find out our answer

P(No Collision) = P (The bugs move in clockwise direction) + P(The bugs move in anti-clockwise direction)
= 0.5 * 0.5 * 0.5 + 0.5 * 0.5 * 0.5
= 0.25

Thus the probability that the bugs won't collide into each other is 0.25 or 1/4.

Solution:

The first number divisible by 7 is 105 and the last is 994 (we have to find the sum of only three digit numbers)

The divisible numbers are 105, 112, 119, ..., 994
Sum of digits = 105 + 112 + 119, ..., + 994

Now if you see the above series it is an arithmetic progression
Where a = 105, d = 7, I = 994

Thus
994 = 105 + (n - 1)*7
or 889 = (n-1)*7
or 127 = n - 1
or n = 128
so sum = (n/2)*(a + l)
= (128/2)*(105 + 994)
= 70336

Solution:

1000

Explanation:
Total number of methods selecting three question out of five in one section = 5*4/2*1 = 10
Since there are three sections and selection has to be made in each of them
Total methods = 10*10*10 = 1000

Solution:

50%

Explanation:
When the coins are tossed there can be eight possible outcomes (2*2*2)
HHH
HTT
HTH
HHT
TTT
THH
THT
TTH

If you find out, we have only four favorable outcomes. Also, the events here are independent and thus we can calculate the probability by using these only.

Thus the probability is 50 percent.

Solution:

What must be kept in mind before you start calculating through any method is that the weights can be put on both the pans.

So the weights - 1, 3, 7, 12, 43, 76, 102

The oldest man that can be weighed using these weights is a man aged 122 years