What is the most efficient/correct approach to this combinatorics question?

Asked Oct 31 '22 at 12:43

Active Oct 31 '22 at 12:52

Viewed 50 times

there are 3 different kinds of textbooks. We have 10 of each textbook. How many ways can we pick 10 textbooks from these 30? (Order of picking doesn't matter)

Additional question: What if we only had 9 of each kind of textbook?

I basically tried writing out every single combination and found that if there are 10 of textbook A, there's only 1 combination. If there are 9, there's 2 combinations.... all the way till theres 0 textbook A. So my final answer is 1+2+3...+11. is this correct and is there a better approach?

I'm quite unsure of my approach and answer because its a combinations question but i didnt use any C or P functions

edited Oct 31 '22 at 12:52

asked Oct 31 '22 at 12:43

bugger

Consider the number of solutions to $$x_1 + x_2 + x_3 = 10 ~: ~x_1,x_2,x_3 \in \Bbb{Z_{\geq 0}}.$$ This is a Stars and Bars problem. For Stars and Bars theory, see this article and this article. – user2661923 Oct 31 '22 at 13:04
Re previous comment, the 2nd question posed is equivalent to adding an upper bound: $$x_1,x_2,x_3 \leq 9.$$ After completely digesting my previous comment, to attack the 2nd question, see this answer. – user2661923 Oct 31 '22 at 13:06

What is the most efficient/correct approach to this combinatorics question?

0 Answers0