Find the coefficient of $$x^5y^3z^{13}$$ in $$(2x+3y-z)^{21}$$

Hi, Shraddha

Let's take 2x=$$t_{1}$$ ; 3y=$$t_{2}$$ and (-z)=$$t_{3}$$ hence, polynomial will be $$(t_{1}+t_{2}+t_{3})^{21}$$ Now as we know that for $$x^{5}$$ $$y^{3}$$ and $$z^{13}$$ , $$(t1)^{5}$$ $$(t2)^{3}$$ and $$(t3)^{13}$$ will be there. So ways of distributing 21 things among 5 , 3 and 13 things will be $$ \frac{21!}{5!3!13!}$$ so this will be coefficient of $$(t1)^{5}$$ $$(t2)^{3}$$ and $$(t3)^{13}$$ hence, corresponding coefficient of x , y and z will be multiplying it by $$2^{5}$$ , $$3^{3}$$ and $$(-1)^{13}$$ so actual coefficient will be = $$\frac{21!}{5!3!13!} \times (2^{5})(3^{3})((-1)^{13})$$. Hope it helps if you want further detailed solutions please let us know.