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