The co-efficient of x in the expansion of `(1-2x^3+3x^5)(1+1/x)^8` is:

Explanation

Suppose `x` occurs at the `(r+1)` th term in the given expression.

Then, we have:

`(1-2 x^{3}+3 x^{5})(1+frac{1}{x})^{8} `

`=(1-2 x^{3}+3 x^{5})(\ ^{8} C_{0}+\ ^{8} C_{1}(frac{1}{x})+\ ^{8} C_{2}(frac{1}{x})^{2}`
`+\ ^{8} C_{3}(frac{1}{x})^{3}+\ ^{8} C_{4}(frac{1}{x})^{4}+\ ^{8} C_{5}(frac{1}{x})^{5}`
`+\ ^{8} C_{6}(frac{1}{x})^{6}+\ ^{8} C_{7}(frac{1}{x})^{7}+\ ^{8} C_{8}(frac{1}{x})^{8}) `

x occurs in the above expresssion at `-2 x^{38}\ ^{8} C_{2}(frac{1}{x^{2}})+3 x^{5} cdot\ ^{8} C_{4}(frac{1}{x})^{4}`

therefore Coefficient of ` x=-2(frac{8 !}{2 ! 6 !})+3(frac{8 !}{4 ! 4 !})=-56+210=154`