How many different ways are there to choose 10 donuts from 20 varieties if...?

Question

• A) At least 6 glazed donuts are chosen?
• B) At most 4 chocolate donuts are chosen?

For Part A, I think I have the correct understanding that it is combinations with repetitions so you take 6 glazed donuts and keep them so you now choose from 4 donuts of 20 varieties. I believe the solution to be C(20+4-1, 4) = C(23,4) if that could be verified that would be appreciated.

For Part B, I am not sure where to start.

Answer 1

How many different ways are there to choose $10$ donuts from $20$ varieties if at least $6$ glazed donuts are chosen?

Your answer to this question is correct.

How many different ways are there to choose $10$ donuts from $20$ varieties if at most $4$ chocolate donuts are chosen?

Hint: Subtract the number of ways the customer could select at least five chocolate donuts from the total number of selections.

Answer 2

Yes, after setting aside the 6 glazed donuts, we indeed just need to figure out how many ways we can get 4 donuts from 20 varieties. This we can do by stars and bars. Use 19 bars to seperate the 20 varieties, and four stars for the donuts to be distributed among the 20 thusly created 'bins', and so you indeed get $23 \choose 4$ possibilities for A.