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May 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 beautiful problem form ISI MStat 2017 PSB Problem 3, where we use the basics of Bijection principle and Vandermone's identity to solve this problem.

Problem

Consider an urn containing 5 red, 5 black, and 10 white balls. If balls are drawn without replacement from the urn, calculate the probability that in the first 7 draws, at least one ball of each color is drawn.

Prerequisites

  • Algebrify the problem
  • Vandermone's Identity $${{n_{1}+\cdots+n_{p}} \choose m} = \sum_{k_{1}+\cdots+k_{p}=m} {n_{1} \choose k_{1}} \times {n_{2} \choose k_{2}} \times \cdots \times {n_{p} \choose k_{p}}$$

Solution

It may give you the intuition, there is atleast in the problem, so let's do complementing counting. Okay! I suggest you to travel that path to help you realize the complexity of that approach.

Nevertheless, let's algebrify the problem.

You want to select \(x_1\) red balls, \(x_2\) blue balls, and \(x_3\) white balls so that \(x_1 + x_2 + x_3 = 7\).

Now, \( 1 \leq x_1 \leq 5\), \(1 \leq x_2 \leq 5\) and \(1 \leq x_3 \leq 10\) represents our desired scenario.

\( 0 \leq x_1 \leq 5\), \(0 \leq x_2 \leq 5\) and \( 0 \leq x_3 \leq 10\) denotes total number of cases.

Now, for each such triplet (\(x_1, x_2, x_3\)) of the number of balls of each colour we have selected, we can select them in \( { 5 \choose x_1} \times {5 \choose x_2 } \times {10 \choose x_3} \).

Let \( P = \{ (x_1, x_2, x_3) | x_1 + x_2 + x_3 = 7, 0 \leq x_1 \leq 5, 0 \leq x_2 \leq 5, 0 \leq x_3 \leq 10 \} \).

Total Number of Ways we can select the 7 balls =

$$ \sum_{(x_1, x_2, x_3) \in P} { 5 \choose x_1} \times {5 \choose x_2 } \times {10 \choose x_3} = { {5 + 5 + 10} \choose { x_1 + x_2+ x_3} } = { {5 + 5 + 10} \choose {7} }$$

\( Q = \{ (x_1, x_2, x_3) | x_1 + x_2 + x_3 = 7, 1 \leq x_1 \leq 5, 1 \leq x_2 \leq 5, 1 \leq x_3 \leq 10 \} \).

Total Number of Ways we can select the 7 balls such that at least one ball of each color is drawn =

$$ \sum_{(x_1, x_2, x_3) \in Q} { 5 \choose x_1} \times {5 \choose x_2 } \times {10 \choose x_3}$$

Let, \( R = \{ (z_1, z_2, z_3) | z_1 + z_2 + z_3 = 4, 0 \leq z_1 \leq 4, 0 \leq z_2 \leq 4, 0 \leq z_3 \leq 9 \} \).

Observe that \(Q\) has a bijection with \(R\). \( x_i = z_i +1\).

Total Number of Ways we can select the 7 balls such that at least one ball of each color is drawn =

$$ \sum_{(x_1, x_2, x_3) \in R} { 4 \choose z_1} \times {4 \choose z_2 } \times {9 \choose z_3} = { {4 + 4 + 9} \choose { z_1 + z_2+ z_3} } = { {4 + 4 + 9} \choose {4} }$$

Exercises

  • Prove the Vandermone's Identity.
  • Find the Probability.
  • Generalize the problem to general numbers and prove it.
  • Generalize the problem where the lower bounds of \(x_i\) are \(m_i\).
  • What if there are upper bounds on \((x_1, x_2, x_3)\)? [ Hint: Generating Functions ].

Stay Tuned! Stay Blessed!

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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