There are \(n\) (distinct) pairs of gloves. In how many ways can \(n\) people select a left handed and right handed glove, such that they do not select a pair? Please give me an expression in terms of \(n\).

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TopNewestThere are \( n! \) ways for the people to first select a left-hand glove. Selecting a right hand glove is then equivalent to finding the number of derangements. – Daniel Wang · 4 years ago

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– Leonardo Cidrão · 4 years ago

But I think it's n! for lefthands and n! for the rights, in total of (n!)^2, isn' it?Log in to reply

– Jimmy Kariznov · 4 years ago

Except each person must have a left hand glove and a right hand glove that aren't a pair.Log in to reply

– Leonardo Cidrão · 4 years ago

I forgot that fact, thanks Jimmy. So I think it'll be (n!)^2 -n!. Is this the correct answer?Log in to reply

Euler's number) and round it to the nearest integer, then you get \(!n\), for all positive integers \(n\). – Tim Vermeulen · 4 years ago

No, that doesn't make sense. There aren't exactly \(n!\) ways for the people to select gloves such that at least one of them has a pair. Like Daniel commented, there are \(n!\) ways to select the left-hand gloves, and the number of ways to select the right-hand gloves is the number of derangements of \(n\) elements, which is denoted by \(!n\). It turns out, if you take \[\frac{n!}{e}\] (where \(e\) isLog in to reply

– Leonardo Cidrão · 4 years ago

Sorry man. I did not study that part of combinatorics yet. Probably I'll see that soon. Thanks for explaning it for me.Log in to reply

– Tim Vermeulen · 4 years ago

That's okay :) I actually didn't know about the number of derangements before I did research after seeing this question.Log in to reply

Your question is incomplete. You are not mentioning whether each person takes equal no(2) of gloves or anyone can select any no. of gloves. If there is no constraint on the no. of gloves a person can choose , then ans. is \( \binom{n}{2}^2 \) – Jatin Yadav · 4 years ago

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Besides, even with your interpretation, for \(n=2\) it's wrong; there are \(2\) ways (a person chooses one left glove and the right glove that doesn't make a pair and the other person takes the rest; there are two such pairs) while your formula gives \(1\). – Ivan Koswara · 4 years ago

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