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

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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. --Jayron32 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 ${\tfrac {5\cdot 4\cdot 3\cdot 2\cdot 1}{(2\cdot 1)\cdot (3\cdot 2\cdot 1)}}$ 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. --Jayron32 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)