Assue a student has 13 writing utensils, 7 pencils, 4 ballpoint pens, and 2 felt pens how many ways can the student choose 9 untensils if at most one ballpoint pen is used?

he has 13 utensils and he has to use 9 utensils and also one ball pen . So he has
13.12.11.4.3.2.1=51184

41184

how many ways can the student choose 9 untensils if at most one ballpoint pen is used?

This means you can have 9 utensils other than ballpoint pens (you pick 9 from the 7 pencils and 2 felt pens) OR 8 utensils other than ballpoint pens and one ballpoint pen from the 4 available.

This doesn't give 41184.


Also, there can't be that much ways, since order is not important here. Taking a blue pencil and a red pencil is the same as a red pencil and a blue pencil.