+26367 How many terms are in the expansion of \(\left(a + b + c + d\right)^5\)?
The multinomial in general is: \(\left( x_1+x_2+\cdots +x_m \right)^n\)
The number of terms in a multinomial sum is \(\#_{n,m}\),
\(\#_{n,m} = \dbinom{n+m-1}{m-1}\)
Source: https://en.wikipedia.org/wiki/Multinomial_theorem
\(\text{Let $m=4$ and $n=5$}\)
\(\begin{array}{|rcll|} \hline \mathbf{\#_{5,4}} &=& \dbinom{5+4-1}{4-1} \\\\ &=& \dbinom{8}{3} \\\\ &=& \dfrac{8}{3}\times \dfrac{7}{2}\times \dfrac{6}{1} \\\\ &=& 8\times 7 \\\\ &=& \mathbf{56} \\ \hline \end{array} \)
In the expansion of \(\left(a + b + c + d\right)^5\) are 56 terms.
