Question

In the expansion of \(\rm\left({x\over2}-{3\over x^2}\right)^{10}\) , find out the coefficient of x⁴ 
1. \(405\over256\)
2. \(504\over259\)
3. \(450\over263\)
4. None of the above

Answer 1

Correct Answer - Option 1 : \(405\over256\)

Concept:

The binomial expansion of (a + b)ⁿ is

(a + b)n = \(\rm \sum_0^n {^nC_r a^rb^{n-r}}\)

Where any r^th term T = \(\rm {^nC_r a^rb^{n-r}}\)

Calculation:

The expansion of \(\rm\left({x\over2}-{3\over x^2}\right)^{10}\) = \(\rm \sum_0^{10} {^{10}C_r \left({x\over2}\right)^{r}\left({-3\over x^2}\right)^{10-r}}\)

The r^th term will be T_r = \(\rm {^{10}C_r \left({x\over2}\right)^{r}\left({-3\over x^2}\right)^{10-r}}\)

T_r = \(\rm {^{10}C_r \left({-3^{10-r}\over 2^{r}}\right)x^{r-2(10-r)}}\)

For the term has x⁴,

r - 2(10 - r) = 4

r - 20 + 2r = 4

3r = 24

r = 8

Now the coefficient of x⁴ is 

C₄ = \(\rm {^{10}C_r \left({-3^{10-r}\over 2^{r}}\right)}\)

Putting r = 8

C₄ = \(\rm {^{10}C_8 \left({-3^{10-8}\over 2^{8}}\right)}\)

C4​ = \(\rm {45 \left({9\over256}\right)}\)

C4​ = \(\boldsymbol{405\over 256}\)