How to get to the powers

Question

Consider: $(a + b + c + d)^{21}$

What will be the coefficient of $a^5b^6c^5d^5$?

Can you also give the general way of finding out coefficients of such terms?

Answer 1

$a^5b^6c^5d^5$ corresponds to the coefficient $$\frac{21!}{5!6!5!5!}$$ in the multinomial $(a+b+c+d)^{21}$. The corresponding value is $\color{red}{41\space 064\space 607\space 584}$.