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Question: Find the number of terms, after simplifying, in $$(x + y + \frac{1}{x} + \frac{1}{y})^{16}$$

I was unable to find any approach to this question. Fully expanding the expression isn't a viable solution. A hint to start the problem would be appreciated.

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HINT:

$$x+y+\frac1x+\frac1y=\frac{(x+y)(1+xy)}{xy}$$

Now, $(a+b)^n$ has $n+1$ terms for integer $n\ge0$

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  • $\begingroup$ Didn't you miss some exponents ? $\endgroup$ – Claude Leibovici Dec 6 '14 at 7:21
  • $\begingroup$ @ClaudeLeibovici, I meant the current version. Thanks for your observation $\endgroup$ – lab bhattacharjee Dec 6 '14 at 7:22
  • $\begingroup$ Well, that has certainly made it easier. Let me try it now. $\endgroup$ – Gummy bears Dec 6 '14 at 7:23
  • $\begingroup$ Well after distributing the exponents, you get 17 terms from each bracket from the numerator. As they are being multiplied together, we will get 17*17 total terms. Is that correct? $\endgroup$ – Gummy bears Dec 6 '14 at 7:26
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    $\begingroup$ @Gummybears. When you see problems like this one, be sure that there is a trick. Consider you are lucky with the power $16$; they could have asked that question with power $12345$ ! $\endgroup$ – Claude Leibovici Dec 6 '14 at 7:40

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