This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 3 based on Counting principle . Let's give it a try !!
Basic Counting Principle
(a) To find the number of integers with non-decreasing digits we will go by this way
First see that the position of given digits are fixed as they have to form non-decreasing digits what can be do is to select number of times the particular digit occurs .
For example 223556 in an integer With non-decreasing digits so here 2 occurs
2 times, 3 occurs 1 times, 4 doesn't occurs any times, 5 occurs 2 times, 6 occur 1 time and finally 7 which doesn't occur any number of times.
So, here we have (0,1,2) possible choices of occurrence of 2 ,(0,1,2) possible choices of occurrence of \( 3, \cdots,\) (0,1) possible choices of occurrence of 7 .
Hence, number of such integers \(=3 \times 3 \times 2 \times 4 \times 2 \times 2-1\).
We are subtracting 1 to exclude the case where no digits has occur any number of times.
(b) To find the number of integers with increasing digits we can go by above method. Here there is only one restriction that a digit must be greater than it's preceding digits.so, no consecutive digits can be equal. Hence every digits has two choices (0,1) of occurrence . Therefore number of such integers = \( 2^{6}-1\) .
We are subtracting 1 to exclude the case where no digits has occur any number of times.
A number is chosen randomly from all the 5 digited numbers.Find out the probability that the digits form a non decreasing sequence.
Hint 1 : Find what is invariant here .
Hint 2 : Use the non-negative integer solutions of a equation formula .
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.
