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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 basic probability of functions. We provide sequential hints so that you can try .

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

But try the problem first…

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

Source

ISI MStat 2018 PSA Problem 13

A First Course in Probability by Sheldon Ross

## Try with Hints

First hint

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 .

Second Hint

$$f(i) = a_i$$

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 $${n \choose m}$$ .

Final Step

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