How many natural numbers less than 1000 have the sum of their digits equal to 5? A.56 B.54 C.26 D.21 E.18

Question

I'm not sure what this problem falls under exactly. It looks like permutation, or maybe exclusion inclusion principle. Anyway, the question is multiple choice, so that should narrow it down.

Answer 1

Start by listing the distinct groupings of three non-negative integers whose sum is $5$: $$1,1,3$$ $$1,2,2$$ $$...$$ $$0,0,5$$

Then for each group figure out how many numbers less that $1000$ contain just those integers.

Answer 2

For single digit answer : 1 only 5

For two digit : we know 1+4=5,2+3=5,5+0=5 total : 5

Three digit : _ _ _ three places

{1,2,3,4,0,5} it's an arrangement of these number's in the above three places

Only option for 5 is 500

for 4, 140,410,401,104

for 3, 302,320,311,131,113,311,230,

for 2, 221,212,122

So the answer is 21