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# 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 from ISI MStat 2018 PSA problem 13 based on probability of functions. We provide sequential hints so that you can try .

## Probability - ISI MStat Year 2018 PSA Problem 13

Consider the set of all functions from ${1,2, \ldots, m}$ to ${1,2, \ldots, n}$ where $n>m .$ If a function is chosen from this set at random, what is the probability that it will be strictly increasing?

• ${n \choose m} / m^{n}$
• ${n \choose m} / n^{m}$
• ${{m+n-1} \choose m} / m^{n}$
• ${{m+n-1} \choose m-1} / n^{m}$

### Key Concepts

combination

Answer: is ${n \choose m} / n^{m}$

ISI MStat 2018 PSA Problem 13

A First Course in Probability by Sheldon Ross

## Try with Hints

What is the total number of functions from ${1,2, \ldots, m}$ to ${1,2, \ldots, n}$ where (n>m)

You have to choose $m$ numbers among ${1,2, \ldots, n}$ and assign it to the ${1,2, \ldots, m}$
For each element of ${1,2, \ldots, m}$, there are $n$ options from ${1,2, \ldots, n}$.
Hence $n^m$ number of functions .

$f(i) = a_i$

The number of ways to select Select $m$ elements among ${1,2, \ldots, n}$ is the same as the number of strictly ascending subsequences of length m taken from 1, 2, 3, ..., n, which is the same as the number of subsets of size m taken from {1,2,3,…,n}, which is ${n \choose m}$ .

Hence the probability that it will be strictly increasing $\frac{ {n \choose m} }{ n^{m} }$

## 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

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