Question

You have `36` cookies. They can be put into packs of `18` or packs of `6` or kept as single cookies. How many different ways are there to package these `36` cookies?

Answer 1

See a solution process below:

Explanation:

Start with what I think is the easiest - packs of 18:

-1) Two packs of 18
-2) One pack of 18 and three packs of 6
-3) One pack of 18 and 2 packs of 6 and 6 single cookies
-4) One pack of 18 and 1 pack of 6 and 12 single cookies
-5) One pack of 18 and 18 single cookies

Next, do the same with packs of 6:

-6) 6 packs of 6
-7) 5 packs of 6 and 6 single cookies
-8) 4 packs of 6 and 12 single cookies
-9) 3 packs of 6 and 18 single cookies
-10) 2 packs of 6 and 24 single cookies
-11) 1 pack of 6 and 30 single cookies

And finally:

-12) 36 single cookies

Answer 2

If all the cookies need to be either in a pack of 6 or a pack of 18 (no loose, left-over cookies) then there are `color(red)3` ways this could be done.

Explanation:

The 36 cookies could contain either `0, 1, or 2` packs of 18 (with the remainder in packs of 6)