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 .
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?
combination
Answer: is \( {n \choose m} / n^{m} \)
ISI MStat 2018 PSA Problem 13
A First Course in Probability by Sheldon Ross
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 \( \frac{ {n \choose m} }{ n^{m} } \)
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
