How many PINs have a digit sum of 15?

Question

I think I can use stars and bars here to find how many combinations of 4 non-negative integers sum to 15, but how do I then count the number of unacceptable cases- i.e. where one of the integers is greater than 9?

Does this answer your question? How many solutions for an equation with simple restrictions — chematwork, Apr 12 '20 at 13:37
I think I can see that substitutions of $x_1 = y_1 +10$ are necessary giving $\binom{8}{3}$ as the solution where one of the digits has to be less than 10. I am struggling to see how to incorporate this in to the rest of my argument. — JDJDJD92, Apr 12 '20 at 21:32

score 1 · Answer 1 · edited Apr 13 '20 at 21:18

1

Is $592$ correct?

Using $\binom{18}{3}$ for the total number of ways for four integers to sum to $15$ using stars and bars.

Then calculating $\binom{8}{3}$ for the number of ways if a digit is $10$ or more. Then subtracting $4$ lots of this, $1$ for each digit.

edited Apr 13 '20 at 21:18

SARTHAK GUPTA

1,308

answered Apr 12 '20 at 21:39

JDJDJD92

31

Yes, this is correct. – Daniel Mathias Apr 13 '20 at 11:54

How many PINs have a digit sum of 15?

1 Answers1