Question

Find the Coefficient of x^11 in the expansion of (3x + 2x^2)9​

Answer 1

Answer:

2 is its coefficient... and answer

Answer 2

Answer: 314928

Concept: This is the question of the 'Binomial Theorem'. So we will use the concept of expansion of   $(x+a)^{n}$

Given:   $(3x+2x^{2} )^{9}$

To Find: Coefficient of $x^{11}$  the expansion of $(3x+2x^{2} )^{9}$

Solution:

Let (r+1) will be the term of coefficient $x^{11}$ in the expansion of   $(3x+2x^{2} )^{9}$

Now (r+1) term = $9C_{r}$×($(3x)^{9-r}$×$(2x^{2})^r$

                        = $9C_{r}$×$3^{9-r}$×$x^{9-r}$×$2^{r}$×$x^{2r}$

                        =$9C_{r}$×$3^{9-r}$×$2^{r}$×$x^{9+r}$

we have asked the power of x to be 11 so

9+r=11

r=2

means (2+1) or we can say in the '3' term of the expansion will have $x^{11}$

The third term will be obtained by putting the value of r=2

3rd term of the given expansion= $9C_{r}$×$3^{9-r}$×$2^{r}$×$x^{9+r}$

                                                     = $9C_{r}$×$3^{7}$×$2^{2}$×$x^{11}$

Coefficient of $x^{11}$

= $\frac{9!}{2!7!}$×$3^{7}$×$2^{2}$

=$\frac{9*8*7!}{2*7!}$×$3^{7}$×$2^{2}$

=36×2187×4

=314928

Hence the coefficient of $x^{11}$ is 314928