ISI MStat PSA 2019 Problem 6 | Basic Counting principles

This is a beautiful problem from ISI MSTAT 2019 PSA problem 6 based on basic counting principles. We provide sequential hints so that you can try.

Basic Counting Principles - ISI MStat Year 2019 PSA - 6

How many times does the digit ‘2’ appear in the set of integers \( \{1,2,…,1000\} \)?

  • 590
  • 600
  • 300
  • 299

Key Concepts

Basic counting principles

Check the Answer

Answer: is 300

ISI MStat 2019 PSA Problem 6

A First Course in Probability by Sheldon Ross

Try with Hints

Let’s count the number of times 2 occurs once.

_ _ _

The position of 2 can be selected in 3 ways. The rest in 9 x 9.

Total number of 2 = 1 x 3 x 9 x 9 = 243

Let’s count the number of times 2 occurs twice.

_ _ _

The positions of 2 can be selected in 3 ways. The rest in 9.

Total number of 2 = 2 x 3 x 9  = 54

Let’s count the number of times 2 occurs thrice.

The positions of 2 can be selected in 1 way. The rest in 1.

Total number of 2 = 3 x 1 x 1  = 3

Total Number of Such Numbers = 243 + 54 + 3 = 300 

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Ordered triples | PRMO 2017 | Question 21

Try this beautiful problem from the Pre-RMO, 2017 based on Ordered triples.

Ordered Triples - PRMO 2017

What is the number of triples (a,b,c) of positive integers such that abc=108?

  • is 107
  • is 60
  • is 840
  • cannot be determined from the given information

Key Concepts

Largest number of triples

Combinatrics

Integers

Check the Answer

Answer: is 60.

PRMO, 2017, Question 21

Elementary Number Theory by David Burton

Try with Hints

abc=\(3^{3}2^{2}\)

a=\(3^{\alpha_1}2^{\beta_1}\), b=\(3^{\alpha_2}2^{\beta_2}\), c=\(3^{\alpha_3}2^{\beta_3}\)

\({\alpha_1}+{\alpha_2}+{\alpha_3}=3\), \({\beta_1}+{\beta_2}+{\beta_3}=2\)

\({5 \choose 2}\), \({4 \choose 2}\)

total= \({5 \choose 2} \times {4 \choose 2}\)=(10)(6)=60 ways.

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