Combination of chips

Question

Mary baked a dozen cookies. She put 45 chips into the batter. each cookie had at least 3 chips but no more than 5 chips. What are all the different possible combinations of chips?

Answer 1

I AM GOING TO TRY TO SOLVE THIS FOR EXACTLY TWELVE COOKIES.

 

Basically 3 3 3 3 3 3 3 3 3 3 3 3 consumes 36 chips (12 cookies) and you'd have 9 more to distribute to some of those.

This could give 3 3 3 3 3 3 3 3 3 3 3 12 as one other way, or

3 3 3 3 3 3 3 3 3 3 4 11

etc. Fairly tedious to continue in this way because there are so many ways to take 9 extra chips and break that down into 1 2 2 2 2  or 1 1 1 2 2 2 etc. etc. (I'm thinking out loud here.)

 

So let's list, systematically, the different ways of dealing with 9 extra chips. Here are the different breakdowns:

9

 

8 1

 

7 2

7 1 1

 

6 3

6 2 1

6 1 1 1

 

5 4

5 3 1

5 2 2

5 2 1 1

5 1 1 1 1  

 

4 3 2

4 3 1 1

4 2 2 1

4 2 1 1 1

4 1 1 1 1 1

 

3 2 1 1 1 1 

3 1 1 1 1 1 1 

 

2 1 1 1 1 1 1 1 

1 1 1 1 1 1 1 1 1 

 

Check this list to see if any other possibilities exist. When the list is finalized, count how many ways are listed here--that's the answer to the question, i.e. how to distribute extras amongst the 12 cookies each originally furnished with 3 chips.