Hello,
In the expansion of (x1 + x2 + x3)^10, what is the coefficient of the x^22?
How we can solve this kind of problems can you explain me?
Thank you.
just to be clear do you mean
$\displaystyle (x + x^2 + x^3)^{10}$
what you want is the multinomial theorem Multinomial theorem - Wikipedia, the free encyclopedia
and you'd let $\displaystyle x_1=x^1$, $\displaystyle x_2=x^2$, $\displaystyle x_3=x^3$
I think I would be inclined to do it this way:
$\displaystyle (x^3+ x^2+ x)^{10}= (x^3+ (x^2+ x))^{10}= \sum_{n= 0}^{10} \begin{pmatrix}10 \\ n\end{pmatrix}(x^2+ x)^nx^{30- 3n}$
So that, for every n, we will have $\displaystyle x^{22}$ provided the power of x in $\displaystyle (x^2+ x)^n$ is 30- 3n- 22= 8- 3n. Of course, $\displaystyle (x^2+ x)^n= \sum_{x= 0}^n\begin{pmatrix}n \\ i\end{pmatrix} (x^2)^ix^{n- i}$ so x will have power 8- 3n when 2i+ n- i= i+ n= 8- 3n or i= 8- 4n. The coefficient of that power will be $\displaystyle \begin{pmatrix}n \\ 8- 4n\end{pmatrix}= \frac{n!}{(8- 4n)!(5n- 8)!}$.
Since that is for every n, the coefficient of $\displaystyle x^{22}$ will be $\displaystyle \sum_{n= 0}^{10} \begin{pmatrix}10 \\ n\end{pmatrix}\begin{pmatrix}n \\ 8- 4n\end{pmatrix}= \sum_{n=0}^{10}\frac{10!}{n!(10- n)!}\frac{n!}{(8- 4n)!(5n- 8)!}= \sum_{n=0}^{10}\frac{10!}{(10-n)!(8- 4n)!(5n- 8)!}$