# Question regarding what appears to be an identity

This is an MCQ we were posed in school recently (I hope you don't mind elementary stuff):

What is $(x-a)(x-b)(x-c)...(x-z)$ ? Options:

$0$

$1$

$2$

$(x^n)-(abcdef...z)$

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Are you sure it didn't go up to $x-z$? –  mixedmath Sep 27 '12 at 2:56
@mixedmath Surely it did, or else none of the choices is correct (not that I particularly like the "solution" in that case). –  Austin Mohr Sep 27 '12 at 2:58
Oh yes, it did. I'm so sorry. But the last option was still $x^n - (abc...z)$ AFAIK. I lost the test paper :( –  octatoan Sep 27 '12 at 2:59
How does it matter if it's $n$ or $z$? –  octatoan Sep 27 '12 at 3:01
@SohamChowdhury The fact that $n$ or $z$ actually makes a difference is something you should think about. –  Erick Wong Sep 27 '12 at 3:07

There is a very basic trick to this problem. It all comes down to a single term (if that's the proper word for it...).

The only real hint I can give is $x$ is a letter between $a$ and $z$...

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+1, Very cool! Also, Thank you @DonAntonio for pointing where I went wrong. –  Emmad Kareem Sep 27 '12 at 4:11
Ah, I see... $(x-a)(x-b)..(x-x)..(x-z)$ where $(x-x)$ is 0. Nice! –  octatoan Sep 27 '12 at 4:22

Hint $\$ What is $\rm\ (24-1)(24-2)(24-3)\cdots (24-26)\$ ?

And what is $\rm\,(x_{24}\!-x_1)(x_{24}\!-x_2)(x_{24}\!-x_3)\cdots (x_{24}\!-x_{26})\$ ?

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@Thanks for the hint. Very good point. –  Emmad Kareem Sep 27 '12 at 4:08