Question

In the following expansion, find the indicated coefficient.

x⁹ in `(1/x + x^2)^18`

Answer 1

Here, a = `1/x`, b = x², n = 18

We have, t_r+1 = ⁿC_r a^n–r .b^r 

= `""^18"C"_"r"(1/x)^(18-"r") (x^2)^"r"`

= ¹⁸C_r (x^–1)^18–r .x^2r

= ¹⁸C_r x^r–18.x^2r

= ¹⁸C_r x^3r–18

To get the coefficient of x⁹, we must have

x^3r–18 = x⁹

∴ 3r – 18 = 9

∴ 3r = 27

∴ r = 9

∴ Coefficient of x⁹

= `""^18"C"_9`

= `(18!)/(9!9!)`

= `(18 xx 17 xx 16 xx 15 xx 14 xx 13 xx 12 xx 11 xx 10)/(9 xx 8 xx 7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1)`

= 48620

∴ Coefficient of x⁹ is 48620