Question

The coefficient of $x^4$in the expansion of ${(1+x+x)}^{10}$ is

Answer 1

Determining the coefficients of $x^4$.

We have the given equation: ${(1+x+x)}^{10}$

We know that in general terms of the given expression is expressed as: $\frac{10!}{p!q!r!}{x}^{q+2r}$

Since the power of $x$ is $4$, therefore $q+2r=4$. [from the general expression.]

Since the value of $p+q+r=10$, we can have the following set of arrangements:

$$\begin{gathered} \text{For}p=6,q=4,r=0, \\ \text{coefficient}=\frac{10!}{(6!\times 4!)} \\ =210 \\ \text{For}p=7,q=2,r=1 \\ \text{coefficient}=\frac{10!}{(7!\times 2!\times 1!)} \\ =360 \\ \text{For}p=8,q=0,r=2 \\ \text{coefficient}=\frac{10!}{(8!\times 2!)} \\ =45 \end{gathered}$$

Therefore, Submission of all the coefficients : $=210+360+45$

$=615$

Hence, the coefficient of $x^4$ is $615$