Question

By the expression \(\left(\frac{x}{3} + \frac{1}{x}\right)^5\), evaluate the middle term in the expression.

(a) 10*(x^5)

(b) \(\frac{1}{5}*(\frac{x}{4})\)

(c) 10*(\(\frac{x}{3}\))

(d) 6*(x^3)

I got this question in an interview for internship.

My question is taken from Counting topic in division Counting of Discrete Mathematics

Answer 1

Correct choice is (c) 10*(\(\frac{x}{3}\))

To elaborate: By using Binomial theorem,the expression \(\left(\frac{x}{3} + \frac{1}{x}\right)^5\) can be expanded as \(\left(\frac{x}{3} + \frac{1}{x}\right)^5 = ^5C_0(\frac{x}{3})^5 + ^5C_1(\frac{x}{3})^4(\frac{1}{x})^1 + ^5C_2(\frac{x}{3})^3(\frac{1}{x})^2\)

\(+ ^5C_3(\frac{x}{3})^2(\frac{1}{x})^3 + ^5C_4(\frac{x}{3})^1(\frac{1}{x})^4 \)

= \((\frac{x}{3})^5 + 5.(\frac{x}{3}) + 10.(\frac{x}{3}) + 10.(\frac{1}{3x}) + 5(\frac{1}{3x^3})\). Hence, the middle term is 10*(\(\frac{x}{3}\)).