the opposite of choose

Question

I remember some of my math related to combinations, and that if I have 6 objects and want to count the number of pairs that I can make from these it is 6 choose 2 which equals 15.

However, now I know that I have 15 pairs and I want to know how many unique objects I need to make these pairs?

Is there a formula for this?

thanks

John Fernley · Answer 1 · 2014-06-26T14:42:33.697

6

Yes, you're asking to solve

$$\binom{n}{2}=k$$

but

$$\binom{n}{2}=\frac{n(n-1)}{2}$$

so

$$n^2 -n-2k=0$$

and

$$n=\frac{1 + \sqrt{1+8k}}{2}$$

edited Jun 26 '14 at 14:42

answered Jun 26 '14 at 14:15

John Fernley

1,218

1

I'll leave you to worry about when $1+4k$ is an odd square, I don't feel it's what you wanted to hear here – John Fernley Jun 26 '14 at 14:19
Hmmmm, $\sqrt{61}$ is not quite integer... You should change $\sqrt{1+4k}$ to $\sqrt{1+8k}$... – barak manos Jun 26 '14 at 14:32
You are indeed right, thanks – John Fernley Jun 26 '14 at 14:43
great thanks! I guess I could have worked that out. – user160254 Jun 27 '14 at 07:31
Yeah, just try to not think of factorials as anything mysterious – John Fernley Jun 27 '14 at 22:30

the opposite of choose

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