The digit sum of a non-negative integer to be the sum of its digits.

For example, the digit sum of $$1+2+3=6$$

a) How many positive integers less than 100 have a digit sum equal to $$8$$?

Let n be a positive integer less than $$10$$

b) How many positive integers less than 100 have a digit sum equal to $$n$$?

c) How many positive integers less than 1000 have a digit sum equal to $$n$$?

d) How many positive integers from $$500$$-$$999$$ have a digit sum equal to $$8$$?

e) How many positve integers less than $$1000$$ have a digit sum equal to $$8$$, and one digit at least $$5$$?

f) What is the total the digit sums of the integers from $$0$$ to $$999$$ inclusive?

a) $$(0,8)(1,7)(2,6)(3,5)$$… $$(6,2)(7,1)(8,0)=9$$ integers

b) $$(0,n)(1, n-1)(2, n-2)(3, n-3)$$… $$(n-1,1)(n,0)= n+1$$ integers

c) $$[(0,0,n)… (0,n,0)] + [(1,0,n-1)… (1,n-1,0)] +…….+[(n,0,0)]$$

$$=[n+1]$$ ways $$+ n$$ ways $$+ [n-1]$$ ways $$+…. +1$$ way

$$=(n+1)(n+2)/2$$ integers

d) $$[(5,0,3)…(5,3,0)] + [(6,0,2)…(6,2,0))] + [(7,0,1)((7,1,0)] + [(8,0,0)]$$

$$= 4 + 3 + 2 + 1 = 10$$ integers

e) $$10$$ ways $$1^{st}$$ digit $$⩾ 5 + 10$$ ways $$2^{nd}$$ digit $$⩾ 5 + 10$$ ways $$3$$ digit $$⩾ 5$$

$$=30$$ integers

f) $$1000$$ numbers implies $$0-9$$ appear in each column $$100$$ times

$$=(0+1+2+3+…+8+9) \times3 \times100=13500$$ total