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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.

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

- 66
- 24
- 28
- 120

Basic counting principles

But try the problem first...

Answer: is

Source

Suggested Reading

ISI MStat 2019 PSA Problem 4

A First Course in Probability by Sheldon Ross

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 .

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