Question

Find the coefficient of x¹¹ in the expansion of `(x^3 - 2/x^2)^12`

Answer 1

Let the general term, i.e., (r + 1)^th contain x¹¹.

We have `"T"_(r + 1) = ""^12"C"_r  (x^3)^(12 - r)  (- 2/x^2)^r`

= `""^12"C"_r  x^(36 - 3r - 2r)  (- 1)^r  2r`

= `""^12"C"_r  (-1)^r  2r  x^(36 - 5r)`

Now for this to contain x¹¹

We observe that 36 – 5r = 11

i.e., r = 5

Thus, the coefficient of x¹¹ is  

`""^12"C"_5  (-1)^5  2^5 = - (12 xx 11 xx 10 xx 9 xx 8)/(5 xx 4 xx 3 xx 2) xx 32`

= – 25344