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  • What is the expansion of  <img alt="\left ( \frac{x}{2}+\frac{2}{y} \right )^4?" src="http://entrancecorner.oncodecogs.com/gif.latex?%5Cleft%20%28%20%5Cfrac%7Bx%7D%7B2%7D&plus;%5Cfrac

What is the expansion of  \left ( \frac{x}{2}+\frac{2}{y} \right )^4?

Option: 1

\left ( \frac{x}{2}+\frac{2}{y} \right )^4=\\^4C_0\left ( \frac{x}{2} \right )^4+^4C_1\left ( \frac{x}{2} \right )^3\cdot\left ( \frac{2}{y} \right )+^4C_2\left ( \frac{x}{2} \right )^2\cdot\left ( \frac{2}{y} \right )^2+^4C_1\left ( \frac{x}{2} \right )\cdot\left ( \frac{2}{y} \right )^3+^4C_0\left ( \frac{2}{y} \right )^4

                                


Option: 2

\left ( \frac{x}{2}+\frac{2}{y} \right )^4=\\^4C_0\left ( \frac{x}{2} \right )^4+^4C_1\left ( \frac{x}{2} \right )^3\cdot\left ( \frac{2}{y} \right )+^4C_2\left ( \frac{x}{2} \right )^2\cdot\left ( \frac{2}{y} \right )^2+^4C_2\left ( \frac{x}{2} \right )\cdot\left ( \frac{2}{y} \right )^3+^4C_1\left ( \frac{2}{y} \right )^4


Option: 3

\left ( \frac{x}{2}+\frac{2}{y} \right )^4=\\^4C_4\left ( \frac{x}{2} \right )^1+^4C_4\left ( \frac{x}{2} \right )^2\cdot\left ( \frac{2}{y} \right )+^4C_4\left ( \frac{x}{2} \right )^3\cdot\left ( \frac{2}{y} \right )^2+^4C_4\left ( \frac{x}{2} \right )^4\cdot\left ( \frac{2}{y} \right )^3+^4C_4\left ( \frac{2}{y} \right )^4


Option: 4

None of the above


Answers (1)

best_answer

For any natural number n, binomial expansion is

(x+y)^{n}=^{n} C_{0} x^{n}+^{n} C_{1} x^{n-1} y+^{n} C_{2} x^{n-2} y^{2}+\cdots+^{n} C_{n}y^{n}\;\;where,\;n\in \mathbb{N} \\
 

Now,

\left ( \frac{x}{2}+\frac{2}{y} \right )^4=\\^4C_0\left ( \frac{x}{2} \right )^4+^4C_1\left ( \frac{x}{2} \right )^3\cdot\left ( \frac{2}{y} \right )+^4C_2\left ( \frac{x}{2} \right )^2\cdot\left ( \frac{2}{y} \right )^2+^4C_3\left ( \frac{x}{2} \right )\cdot\left ( \frac{2}{y} \right )^3+^4C_4\left ( \frac{2}{y} \right )^4

Now, since

\\^4C_3=\;^4C_1,\;\;\\^4C_4=\;^4C_0

hence option A is correct

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