Maths: Probablity Problem

Two distinct numbers x and y are chosen at random from the first 30 natural numbers. What is the probability that a2 - b2 is divisible by 3?

Well, instead of giving my clumsy solution, I'd provide the one from Karthik.

Hint: 3|(a2 - b2)
=> 3|(a - b)(a + b)
=> 3|(a - b) or 3|(a + b) or both

Find the doublets (a, b) such that either a - b or a + b or both are divisible by 3.

Class 1: {1, 4, 7, ... }, {2, 5, 8, ... }, {3, 6, 9, ... }. Each set has 10 numbers. Total number of such combinations is 3 * (10 * 9) / 2 = 135

Class 2: {1, (2, 5, 8, ..)}, {2, (4, 7, ...)}, ...

Now look at the three sets {1, 2, ... }, {2, 4, ... }, {3, 6, ... }. In these sets, we have covered all numbers in range [1, 30]. Forgetting about the third set, to find two numbers such that their sum is divisible by 3, all we need to do it pick one number from the first set and another from the second.

Total number of combinations = 10 * 10 = 100

Class 3: The combinations of {3, 6, 9, ... } will be repeated in both. So, we need to count them only once. We have already counted in Class 1, so we'll not count again.

Total number of picking two numbers is ³⁰C₂ = 435

Probability =

(135 + 100) / 435
  = 235/435
  = 470/870
  = 47/87

Physics: Body in circular Motion

Scenario: A body is moving in a circular motion with constant speed. A practical example can be -- a car moving in circular motion with constant speed. The surface of motion is horizontal.

Problem: In which direction does the friction force apply? Assume there is only pure rolling at the contact surface.

Solution: Have a look at the following diagram that demonstrates the change in velocity (Borrowed from Physics Classroom).

The magnitues of vi and vf are same and as the points A and B get closer, the direction of dv becomes radial.

Conclusion 1: The net acceleration of the body in this scenario is radial and directed towards the center. Or in other words, the net force on the body is towards the center.

Now, look at the following diagram.

I have not mentioned about gravitational forces and reaction from ground thereof since that is not of any importance here.

• Fn denotes the net force (radial).
• Fm denotes the force by the motor (or engine) of the car.
• Ff denotes the frictional force.

Here's the simple relation between the three forces:
Fn = Fm + Ff
The magnitude of Ff depends on the magnitude of the speed and the force by the engine.

Conclusion 2: The frictional force is oblique to the direction of motion and directed inwards towards the center (not radially).