Question

Question 5# (Combinations, including stars and bars') Problem: How many PINs have digit suam 20? (A PIN is string abcd of 4 decimal digits eg 6806) As a first attempt at answering this: (a) How many different solutions in non-negative integers has the equation a+b+c+d-20? Hint: 20 stars and 3 bars. The count in (a) is much too big for our problem because it includes many solutions that contain non-decimal digits; ie. values of a, b, c or d that are greater than 9. We need to climinate these. For example: (b) How many of the solutions to a+btctd-20 have one of the 'digits' equal to 15? Hint: In how many places can the 15 occur, and what's left when you remove it? (e) How many of the solutions to a+bictd 20 have one or two of the 'digits' equal to 10 (d) Given that (2( mnathematical induction, use all the previous answers to calculate the answer to the original problem. As a check, you should find that the answer has digit sum 12 for any n> 1 (this can be proved by

Answer 1

as iutw KI o) atbtcd 20 hnine indega 2.

o , o, lo, 0 1. 2