This is a beautiful problem from ISI MSTAT 2019 PSA problem 4 based on basic counting principles. We provide sequential hints so that you can try.

Basic Counting Principles – ISI MStat 2019 PSA – 4


What is the number of 6 digit positive integers in which the sum of the digits is at least 52?

  • 66
  • 24
  • 28
  • 120

Key Concepts


Basic counting principles

Check the Answer


But try the problem first…

Answer: is

Source
Suggested Reading

ISI MStat 2019 PSA Problem 4

A First Course in Probability by Sheldon Ross

Try with Hints


First hint

Find the Minimum Digit for each case of sum of the digits (S).

S = 54, Minimum Digit = 9

S = 53, Minimum Digit = 8

S = 54, Minimum Digit = 7 or 8

Second Hint

Let’s find the Second Minimum Digit and the Third Minimum Digit  for S = 53 and S = 52.

S = 53,
Second Minimum = 9

S = 52,
Minimum Digit = 7,
Second Minimum = 9

S = 52,
Minimum Digit = 8,
Second Minimum = 8,
Third Minimum = 9

Final Step

Now it’s time for counting

S = 54
{999999}

S = 53,
{8,9,9,9,9,9} : Total = 6

S = 54,
{7,9,9,9,9,9} : Total = 6
{8,8,9,9,9,9} : Total = 15

Hence in total there are 1+6+6+15=28 such numbers .

Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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