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# ISI MStat 2018 PSA Problem 13 | Probability of functions

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 $\frac{ {n \choose m} }{ n^{m} }$

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