Counting in Geometry

(a) In a plane, if there are n points of which no three are collinear, then

1. The number of straight lines that can be formed by joining them is ⁿC₂.
2. The number of triangles that can be formed by joining them is ⁿC₃.
3. The number of polygons with k sides that can be formed by joining them is ⁿC_k.

(b) In a plane, if there are n points out of which m points are collinear, then

1. The number of straight lines that can be formed by joining them is ⁿC₂ – ^mC₂ + 1.
2. The number of triangles that can be formed by joining them is ⁿC₃ – ^mC₃.
3. The number of polygons with k sides that can be formed by joining them is ⁿC_k – ^mC_k.

(c) The number of diagonals of an n sided polygon is ⁿC₂ – n = n × (n – 3)/2

I am dividing this method into four parts and we will discuss each part one by one:

a. Last two digits of numbers which end in one
b. Last two digits of numbers which end in 3, 7 and 9
c. Last two digits of numbers which end in 2
d. Last two digits of numbers which end in 4, 6 and 8

Before we start, let me mention binomial theorem in brief as we will need it for our calculations.

Last two digits of numbers ending in 1

Let’s start with an example.

What are the last two digits of 31⁷⁸⁶?

The rules of business can be applied to any competitive environment. Just as businesses keep trying to achieve an extra edge and sell more and more products so do students preparing for competitive exams such as CAT 2018. The focus should always be on trying to achieve more and more. For CAT aspirants, this is the golden rule- always be focused on your goal. That means giving away all your distractions: that movie which other friends are going to watch, those phone calls, whatsapp, facebook, or twitter, those sms, those days of boredom when you do not want to study, and so much more. For every second, minute, hour or day that you lose, someone is trying to get ahead of you. The fact is that you are not going to become a CAT cracker overnight; you will have to grow inch by inch. And the other fact is that it is easier to increase your percentile to 95 or 97 from 70 or 80 percentile but it is much much tougher to increase it further to 99+. So the more you are at the top, the harder it is to grow further. And it takes focus and dedication climb day by day.

I am very fond of today’s chapter. These tricks are used by me off an on and I teach them in my first class also. The result is that all my students are able to work out calculations mentally by the end of the class. The sad fact is that they do not continue it once they get home, which is a pity. If they keep up the practise for 3 or 4 months, they would see themselves doing wonders, as I saw with myself. But practicing it day in and day out again takes focus and vision of one’s goal.

At every step in life, we face diverse problems. They may be related to our family, health, education, career and much more. There can be no one who says that I never had a problem. Srimad Bhagwatam [10.14.58] says ‘padam padam yad vipadam’ which means that at every step in life we face problems. Now how to eradicate our problems is the biggest problem. One thing is sure that we cannot escape the discomfort created because of the problems of life. So the wiser advice will be to find out the root cause of our problems and try to kill it. And remember, we must help each other by solving each other’s problems. That makes life smoother. You get dependable and caring friends which you’d like to carry along throughout life.

Coming back to our math worries, many students have repeatedly asked us to create a Quant corner where we can post our (beautiful) solutions to beautiful math problems and discuss some small concepts. So here we are! This place is open for discussions as that is the best way of learning. So let’s start today’s mathematical extravaganza.