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# help

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How many terms are in the expansion of (a + b + c + d)^5?

Dec 2, 2019

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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 $$\#_{n,m}$$,

$$\#_{n,m} = \dbinom{n+m-1}{m-1}$$

$$\text{Let m=4 and n=5}$$

$$\begin{array}{|rcll|} \hline \mathbf{\#_{5,4}} &=& \dbinom{5+4-1}{4-1} \\\\ &=& \dbinom{8}{3} \\\\ &=& \dfrac{8}{3}\times \dfrac{7}{2}\times \dfrac{6}{1} \\\\ &=& 8\times 7 \\\\ &=& \mathbf{56} \\ \hline \end{array}$$

In the expansion of $$\left(a + b + c + d\right)^5$$ are 56 terms.

Dec 2, 2019