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 to
where
If a function is chosen from this set at random, what is the probability that it will be strictly increasing?
combination
Answer: is
ISI MStat 2018 PSA Problem 13
A First Course in Probability by Sheldon Ross
What is the total number of functions from to
where (n>m)
You have to choose numbers among
and assign it to the
For each element of , there are
options from
.
Hence number of functions .
The number of ways to select Select elements among
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
.
Hence the probability that it will be strictly increasing
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 to
where
If a function is chosen from this set at random, what is the probability that it will be strictly increasing?
combination
Answer: is
ISI MStat 2018 PSA Problem 13
A First Course in Probability by Sheldon Ross
What is the total number of functions from to
where (n>m)
You have to choose numbers among
and assign it to the
For each element of , there are
options from
.
Hence number of functions .
The number of ways to select Select elements among
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
.
Hence the probability that it will be strictly increasing