Question

Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`

Answer 1

The given expression is `(x^4 - 1/x^3)^15`

General Term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`

= `""^15"C"_r  (x^4)^(15 - r) (- 1/x^3)^r`

= `""^15"C"_r  (x)^(60 - 4r) (-1)^r * 1/x^(3r)`

= `""^15"C"_r (-1)^r * 1/(x^(3r - 60 + 4r))`

= `""^15"C"_r (-1)^r * 1/(x^(7r - 60))`

To find the coefficient of `1/x^17`

Put 7r – 60 = 17

⇒ 7r = 60 + 17

⇒ 7r = 77

∴ r = 11

Putting the value of r in the above expression, we get

= `""^15"C"_11 (-1)^11 * 1/x^17`

= `- ""^15"C"_4 * 1/x^17`

= `- (15 xx 14 xx 13 xx 12)/(4 xx 3 xx 2 xx 1) * 1/x^17`

= `- 1365 * 1/x^17`

Hence, the coefficient of `1/x^17` = – 1365