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Find the coefficient of term independent of x in the expansion of $(1+x+7/x)^7$.

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This is my attempt. I found the number of ways by which we can create a term independent of x.

If we take all the 1's from each bracket, there are $7C7$ ways to do that. Next we take 1 "x"from a bracket and 1 "7/x" from another bracket and take 1 from the remaining brackets, there are $7C1.6C1.7$ ways to do that. Next we take 2"x" from two brackets and 2 "7/x" from another two brackets and take 1 from the remaining brackets, there are $7C2.5C2$ ways to do that. Similarly we take 3"x" from three brackets and 3 "7/x" from other three brackets and take 1 from the remaining brackets, there are $7C3.4C3$ ways to do that.

We add all these 5 terms to get the required coefficient.

But my answer is much lesser than what is given in the textbook.

What case I missed? Did I count wrong?

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  • $\begingroup$ There is apparently a typo in either the problem or the solution. Note it reads "The exponent 11 is to be divided..." implying that the problem they are solving is instead $(1+x+\frac{7}{x})^{11}$ not $(1+x+\frac{7}{x})^7$ $\endgroup$ – JMoravitz Feb 25 '17 at 17:34
  • $\begingroup$ @JMoravitz Yeah that's right $\endgroup$ – Arishta Feb 25 '17 at 17:37
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Your answer of $1+\binom{7}{1}\binom{6}{1}7+\binom{7}{2}\binom{5}{2}7^2+\binom{7}{3}\binom{4}{3}7^3=58605$ is correct for finding the coefficient of the constant term in the expansion of $(1+x+\frac{7}{x})^\color{red}{7}$. wolframalpha

The error is on the books part, either having a typo in the problem statement or having made a mistake in the solution itself. The calculations in the solution are correct for the different problem of finding the coefficient of the constant term in the expansion of $(1+x+\frac{7}{x})^{\color{red}{11}}$ (note the different exponent).

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