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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 $x-z$?2012-09-27
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    @mixedmath Surely it did, or else none of the choices is correct (not that I particularly like the "solution" in that case).2012-09-27
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    Oh yes, it did. I'm so sorry. But the last option was still $x^n - (abc...z)$ AFAIK. I lost the test paper :(2012-09-27
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    How does it matter if it's $n$ or $z$?2012-09-27
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    @SohamChowdhury The fact that $n$ or $z$ actually makes a difference is something you should think about.2012-09-27
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    I've always liked this one.2012-09-27

2 Answers 2

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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.2012-09-27
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    Ah, I see... $(x-a)(x-b)..(x-x)..(x-z)$ where $(x-x)$ is 0. Nice!2012-09-27
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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.2012-09-27