Question

The coefficient $x^{12}$ in ($x^3$ + $x^4$ + $x^5$ +...)${}^3$ ____

• A. 10

Answer 1

The correct option is A 10
We wish to find coefficient of $x^{1}2$ in ($x^3$ + $x^4$ + $x^5$ +...)${}^3$
= ($x^3$ (1 + $x^1$ + $x^2$ + ...))${}^3$
= $x^9$ (1 + x + $x^2$.....)${}^3$

= $\frac{x^9}{(1-x{)}^3}$

= $x^9$ ${\sum }_{r=0}^{\infty }$ ${}^{3-1+r}$$C_r$$X^r$

= $x^9$ ${\sum }_{r=0}^{\infty }$ ${}^{r+2}$$C_r$$X^r$

Now to make $x^{12}$ we need put r = 3
So coefficient to $x^{12}$ is ${}^{3+2}$$C_3$ = ${}^5$$C_3$ = ${}^5$$C_2$ = 10