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Multinomial Expansion

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Question

Find the number of terms in the expansion of ${(2 x + 3 y + z)}^{7}$ .

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Solution

Fact: Number of dissimilar terms in the expansion of $(x_{1} + x_{2} + x_{3} + . . . . . . + x_{r})^{n}$ is $^{n + r - 1} C_{n}$

So in $(2 x + 3 y + z)^{7}$ , we have $r = 3$ and $n = 7$
Thus number of terms in the expansion of $(2 x + 3 y + z)^{7}$ is $^{7 + 3 - 1} C_{7} =^{9} C_{7} = 36$

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