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August 5, 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 is a very beautiful sample problem from ISI MStat PSB 2007 Problem 6 based on counting principle . Let's give it a try !!

Problem- ISI MStat PSB 2007 Problem 6

18 boys and 2 girls are made to stand in a line in a random order. Let \(X\) be the number of boys standing in between the girls. Find
(a) P(X=5)
(b) \( E(X) \)


Prerequisites


Basic Counting Principle

Probability

Discrete random variable

Solution :

If there are j boys in between 2 girls then first we have to choose j boys out of 18 boys in \( {18 \choose j} \) ways now this j boys can arrange among themselves in j! ways and 2 girls can arrange among themselves in 2! ways and now consider these j boys and 2 girls as a single person then this single person along with remaining (18-j) boys can arrange among themselves in (18-j+1)! ways .

Giving all total \( 2! {18 \choose j} j! (18-j+1)! \) possible arrangements .

Again without any restrictions there are (18+2)!=20! arrangements .

Hence \( P(X=j)=\frac{ 2! {18 \choose j} j! (18-j+1)! }{20!} = \frac{19-j}{190} \)

(a) \( P(X=5)=\frac{19-5}{190}=\frac{7}{95} \)

(b) \( E(X)= \sum_{j=0}^{18} \frac{j (19-j )}{190} = 9 \) just computing the sum of first 18 natural number and sum of squares of first 18 natural numbes.


Food For Thought

Find the same under the condition that 18 boys and 2 girls sit in a circular table in a random order .


ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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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?

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