Wikipedia:Reference desk/Archives/Mathematics/2021 October 21

= October 21 =

What's the explanation for this equation?
What's the explanation for this equation? $$\textstyle{5\choose 2}=10$$. As I understand it isn't (5/2), so what's that?--ThePupil (talk) 10:12, 21 October 2021 (UTC)
 * It's a binomial coefficient. Explicitely, $$\textstyle{5\choose 2} = \tfrac{5!}{2! \cdot 3!} = 10$$. --Wrongfilter (talk) 10:35, 21 October 2021 (UTC)
 * It is a common function in combinatorics, and is often read as "5 choose 2", it represents the number of possible ways you can have the number of the bottom item chosen from a group the size of the top item. For example, "5 choose 2" means that if I have a set of 5 items (ABCDE) then there are 10 possible ways that you can pick two from that set (1-AB, 2-AC, 3-AD, 4-AE, 5-BC, 6-BD, 7-BE, 8-CD, 9-CE, 10-DE).  Of course, like much of math, treating it as a binomial coefficient means that it can be generalized as a function to many more applications than merely choosing a smaller group from a larger group.  -- Jayron 32 16:13, 21 October 2021 (UTC)
 * Thank you! Could I look at it as if it's a multiplication? For instance: $$\textstyle{4\choose 2}=8$$, or $$\textstyle{7\choose 4}=28$$ and so on.--ThePupil (talk) 16:18, 21 October 2021 (UTC)
 * No, it means $$\tfrac{5!}{2! \cdot (5-2)!}$$ where ! means factorial, multiply all of the integers up to and including the number. To expand the initial example, it would be $$\tfrac{5\cdot4\cdot3\cdot2\cdot1}{(2\cdot1)\cdot (3\cdot2\cdot1)}$$ which is 120/12 = 10.  The fact that $$\textstyle{5\choose 2}$$ is equal to 5*2 is a coincidence, and would not normally work that way.  For example, $$\textstyle{7\choose 4}$$ is 35, while 7*4 is 28.  -- Jayron 32 16:26, 21 October 2021 (UTC)
 * Thank you for the amazing explanation! You realy made me happy to understand this firstly in my life. (As any time I learn something new in science).--ThePupil (talk) 18:40, 21 October 2021 (UTC)