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?
Basic counting principles
Answer: is
ISI MStat 2019 PSA Problem 4
A First Course in Probability by Sheldon Ross
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
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
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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