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I have a pretty simple straightforward question.

Q) Find the value of $x$ in the following: $$\frac{x}{10!} = \frac{1}{8!} + \frac{1}{9!}$$

Instinctively, I do the quickest thing I know how to do. $$\times10!\implies x = \frac{10!}{8!} + \frac{10!}{9!} = 9 \times 10 + 9 = 99$$ That seemed pretty easy and the back of my textbook says it's the right answer.

Just for fun, I do it again. Except this time, I initially multiply the equation with $9!$ instead of $10!$,

$$\times 9!\implies\frac{9!}{10!}x = \frac{9!}{8!} + \frac{9!}{9!}$$ $$\implies\frac{x}{10} = 9 + 1$$ $$\times 10\implies x = 10\times10 = 100$$ Sure, by multiplying by $10$ in the last step, I've still multiplied by $10!$.
But now, this $1\%$ variation in the answer has popped up.

So, I did what any highschooler would do in this situation;
I consulted Wolfram Alpha which told me that my $2^{\text{nd}}$ trial was correct.

Who should I believe? A computational knowledge engine or the back of my textbook?

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    $\begingroup$ $$\frac{(n+1)!}{n!}=n+1\;\ldots$$ $\endgroup$ – DonAntonio Mar 1 '14 at 17:42
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    $\begingroup$ Let me answer your question: you should believe your brain! $\endgroup$ – user63181 Mar 1 '14 at 17:42
  • $\begingroup$ @Sami: I never thought about that, but I will keep it mind. $\endgroup$ – Nick Mar 1 '14 at 18:02
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    $\begingroup$ @Don: Will recheck my calculations before I post questions again. Sir, yes, sir. $\endgroup$ – Nick Mar 1 '14 at 18:04
  • $\begingroup$ Lessons learnt: (1) I am an idiot who should find his brain and (2) do not doubt Wolfram's calculations. $\endgroup$ – Nick Mar 1 '14 at 18:09
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There's a mistake in your first attempt:

$$ x = \frac{10!}{8!} + \frac{10!}{9!} = 10 \times 9 + 10 = 100 $$

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  • $\begingroup$ hehe, omg, I can't believe i still make stupid mistakes like this. Thanks :D $\endgroup$ – Nick Mar 1 '14 at 17:40
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    $\begingroup$ Even the text book made it. So, it's not just you. :P $\endgroup$ – Priyatham Mar 1 '14 at 17:41
  • $\begingroup$ A better question would be what were the odds of that happening? $\endgroup$ – Nick Mar 1 '14 at 18:00
  • $\begingroup$ @Nick a question: did you look at the answer in the back of the book before or after you solved it? $\endgroup$ – Steven Stadnicki Jun 4 '14 at 1:24
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    $\begingroup$ @Steven: After.That was why it was so shocking. Now looking upon this silliness of the past version of myself, I feel like I ought to meditate more inorder to prevent my regression to idiocy. $\endgroup$ – Nick Jul 20 '14 at 7:46

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