Bars and Stars question

I was doing some practice problems in the stars and bars chapter of the book, and I came across this question:

Screen Shot 2023-02-01 at 6.57.09 PM1179×691 72.7 KB

How is it that {16 \choose 9} accounts for the non-increasing digits portion of this question?

IMG_89521920×868 113 KB

After reading your questions, i do not believe that my answer is what you are looking for. You seem to have a great grasp on the subject with your solutions to the last quiz prep (thanks for that!). I believe you are asking why do we not subtract anything… and i agree with you.

You don’t subtract anything because all the numbers fit the criteria of a non-increasing phone number. It doesn’t say the phone number has to decrease it just can’t increase so having 9999999 is just as valid as 8888888 or any other repeated number like 9888888. Basically, it’s not really a restriction in terms of actually choosing the numbers for each position