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April 23, 2020

How to Pursue Mathematics after High School?

For Students who are passionate for Mathematics and want to pursue it for higher studies in India and abroad.

This problem is a beautiful and simple application of the bijection principle to count the number of non-consecutive selection of integers in combinatorics from Problem 3 of ISI MStat 2019 PSB.

Problem - Non-Consecutive Selection

Elections are to be scheduled for any seven days in April and May. In how many ways can the seven days be chosen such that elections are not scheduled on two consecutive days?

Prerequisites

  • \(a < b\), then \(a\) and \(b+1\) are never consecutive.
  • Combination ( Choose Principle )
  • Bijection Principle

Solution

The problem is based on the first prerequisite mainly. That idea mathematicalizes the problem.

Out of the 61 days in April and May, we have to select 7 non-consecutive days. Let's convert this scenario to numbers.

Out of {\(1, 2, 3, ... , 61\)}, we have to select 7 non-consecutive numbers.

Lemma

\(y_1 < y_2 < y_3 < ... < y_7\) are 7 non-consecutive numbers \( \iff\) \(y_i\) is of the form \( x_i + (i-1) \) where \(x_1 < x_2 < x_3 < ... < x_7\).

For example

You select {1, 3, 4, 5, 6, 8}. You change it to {1, 3 + 1, 4 + 2 , 5 + 3, 6 + 4 , 8 + 5} = { 1, 4, 6, 8, 10 , 13}, which are never consecutive.

Essentially, we are counting the non-consecutive integers in a different way, which helps us to count them.

So, we have to choose \(x_1 < x_2 < x_3 < ... < x_7\), where the maximum \(x_7 + (7-1) = x_7 + 6 \leq 61 \Rightarrow x_7 \leq 55\).

Hence, the problem boiled down to choosing \(x_1 < x_2 < x_3 < ... < x_7\) from {\(1, 2, 3, ... , 55\)}, which is a combination problem.

We can have to just choose 7 such numbers. The number of ways to do so is \( {55}\choose{7}\).

What to do to shape your Career in Mathematics after 12th? 

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

  • What are some of the best colleges for Mathematics that you can aim to apply for after high school?
  • How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
  • What are the best universities for MS, MMath, and Ph.D. Programs in India?
  • What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
  • How can you pursue a Ph.D. in Mathematics outside India?
  • What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Want to Explore Advanced Mathematics at Cheenta?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

To Explore and Experience Advanced Mathematics at Cheenta
Register here

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